id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-14100 | Fourier Sine Coefficients for Odd Function over Symmetric Range | Let $\map f x$ be an odd real function defined on the interval $\openint {-\lambda} \lambda$.
Let the Fourier series of $\map f x$ be expressed as:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
Then for all $n \in \Z_{> 0}$:... | As suggested, let the Fourier series of $\map f x$ be expressed as:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
By definition of Fourier series:
:$b_n = \ds \frac 1 \lambda \int_{-\lambda}^{-\lambda + 2 \lambda} \map f x \... | Let $\map f x$ be an [[Definition:Odd Function|odd]] [[Definition:Real Function|real function]] defined on the [[Definition:Real Interval|interval]] $\openint {-\lambda} \lambda$.
Let the [[Definition:Fourier Series|Fourier series]] of $\map f x$ be expressed as:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = ... | As suggested, let the [[Definition:Fourier Series|Fourier series]] of $\map f x$ be expressed as:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
By definition of [[Definition:Fourier Series|Fourier series]]:
:$b_n = \ds \f... | Fourier Sine Coefficients for Odd Function over Symmetric Range | https://proofwiki.org/wiki/Fourier_Sine_Coefficients_for_Odd_Function_over_Symmetric_Range | https://proofwiki.org/wiki/Fourier_Sine_Coefficients_for_Odd_Function_over_Symmetric_Range | [
"Odd Functions",
"Fourier Series"
] | [
"Definition:Odd Function",
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Fourier Series"
] | [
"Definition:Fourier Series",
"Definition:Fourier Series",
"Sine Function is Odd",
"Odd Function Times Odd Function is Even",
"Definition:Even Function"
] |
proofwiki-14101 | Fourier Series for Odd Function over Symmetric Range | Let $\map f x$ be an odd real function defined on the interval $\openint {-\lambda} \lambda$.
Then the Fourier series of $\map f x$ can be expressed as:
:$\map f x \sim \ds \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$
where for all $n \in \Z_{> 0}$:
:$b_n = \ds \frac 2 \lambda \int_0^\lambda \map f x \... | By definition of the Fourier series for $f$:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
From Fourier Cosine Coefficients for Odd Function over Symmetric Range:
:$a_n = 0$
for all $n \in \Z_{\ge 0}$.
From Fourier Sine Coe... | Let $\map f x$ be an [[Definition:Odd Function|odd]] [[Definition:Real Function|real function]] defined on the [[Definition:Real Interval|interval]] $\openint {-\lambda} \lambda$.
Then the [[Definition:Fourier Series|Fourier series]] of $\map f x$ can be expressed as:
:$\map f x \sim \ds \sum_{n \mathop = 1}^\infty ... | By definition of the [[Definition:Fourier Series|Fourier series]] for $f$:
:$\map f x \sim \dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
From [[Fourier Cosine Coefficients for Odd Function over Symmetric Range]]:
:$a_n = 0$
for all $n ... | Fourier Series for Odd Function over Symmetric Range | https://proofwiki.org/wiki/Fourier_Series_for_Odd_Function_over_Symmetric_Range | https://proofwiki.org/wiki/Fourier_Series_for_Odd_Function_over_Symmetric_Range | [
"Odd Functions",
"Fourier Series"
] | [
"Definition:Odd Function",
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Fourier Series"
] | [
"Definition:Fourier Series",
"Fourier Cosine Coefficients for Odd Function over Symmetric Range",
"Fourier Sine Coefficients for Odd Function over Symmetric Range"
] |
proofwiki-14102 | Half-Range Fourier Sine Series over Negative Range | Let $\map f x$ be a real function defined on the open real interval $\openint 0 \lambda$.
Let $f$ be expressed using the half-range Fourier sine series over $\openint 0 \lambda$:
:$\ds \map S x \sim \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$
where:
:$b_n = \ds \frac 2 \lambda \int_0^\lambda \map f x ... | From Fourier Series for Odd Function over Symmetric Range, $\map S x$ is the Fourier series of an odd real function over the interval $\openint 0 \lambda$.
We have that $\map S x \sim \map f x$ over $\openint 0 \lambda$.
Thus over $\openint {-\lambda} 0$ it follows that:
:$\map S x = -\map f {-x}$
{{qed}} | Let $\map f x$ be a [[Definition:Real Function|real function]] defined on the [[Definition:Open Real Interval|open real interval]] $\openint 0 \lambda$.
Let $f$ be expressed using the [[Definition:Half-Range Fourier Sine Series|half-range Fourier sine series]] over $\openint 0 \lambda$:
:$\ds \map S x \sim \sum_{n \m... | From [[Fourier Series for Odd Function over Symmetric Range]], $\map S x$ is the [[Definition:Fourier Series|Fourier series]] of an [[Definition:Odd Function|odd]] [[Definition:Real Function|real function]] over the [[Definition:Real Interval|interval]] $\openint 0 \lambda$.
We have that $\map S x \sim \map f x$ over ... | Half-Range Fourier Sine Series over Negative Range | https://proofwiki.org/wiki/Half-Range_Fourier_Sine_Series_over_Negative_Range | https://proofwiki.org/wiki/Half-Range_Fourier_Sine_Series_over_Negative_Range | [
"Half-Range Fourier Series"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open",
"Definition:Half-Range Fourier Sine Series",
"Definition:Real Interval",
"Definition:Real Function",
"Definition:Half-Range Fourier Sine Series",
"Definition:Odd Function"
] | [
"Fourier Series for Odd Function over Symmetric Range",
"Definition:Fourier Series",
"Definition:Odd Function",
"Definition:Real Function",
"Definition:Real Interval"
] |
proofwiki-14103 | Half-Range Fourier Sine Series/Cosine over 0 to Pi | :500pxrightthumb$\map f x$ and its $7$th approximation
On the interval $\openint 0 \pi$:
{{begin-eqn}}
{{eqn | l = \cos x
| r = \frac 8 \pi \sum_{m \mathop = 1}^\infty \frac {m \sin 2 m x} {4 m^2 - 1}
| c =
}}
{{eqn | r = \frac 8 \pi \paren {\frac {\sin 2 x} {1 \times 3} + \frac {2 \sin 4 x} {3 \times 5} +... | Let $\map f x$ be the function defined as:
:$\forall x \in \openint 0 \pi: \map f x = \cos x$
Let $f$ be expressed by a half-range Fourier sine series:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$
where for all $n \in \Z_{> 0}$:
:$b_n = \ds \frac 2 \lambda \int_0^\lambda \cos x \sin... | :[[File:Sneddon-1-5-Example4.png|500px|right|thumb|$\map f x$ and its $7$th approximation]]
On the [[Definition:Real Interval|interval]] $\openint 0 \pi$:
{{begin-eqn}}
{{eqn | l = \cos x
| r = \frac 8 \pi \sum_{m \mathop = 1}^\infty \frac {m \sin 2 m x} {4 m^2 - 1}
| c =
}}
{{eqn | r = \frac 8 \pi \pare... | Let $\map f x$ be the [[Definition:Real Function|function]] defined as:
:$\forall x \in \openint 0 \pi: \map f x = \cos x$
Let $f$ be expressed by a [[Definition:Half-Range Fourier Sine Series|half-range Fourier sine series]]:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$
where f... | Half-Range Fourier Sine Series/Cosine over 0 to Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Sine_Series/Cosine_over_0_to_Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Sine_Series/Cosine_over_0_to_Pi | [
"Examples of Half-Range Fourier Series"
] | [
"File:Sneddon-1-5-Example4.png",
"Definition:Real Interval"
] | [
"Definition:Real Function",
"Definition:Half-Range Fourier Sine Series",
"Primitive of Sine of a x by Cosine of a x",
"Sine of Integer Multiple of Pi",
"Primitive of Sine of a x by Cosine of b x",
"Cosine of Zero is One",
"Cosine of Integer Multiple of Pi",
"Cosine of Integer Multiple of Pi"
] |
proofwiki-14104 | Half-Range Fourier Cosine Series over Negative Range | Let $\map f x$ be a real function defined on the open real interval $\openint 0 \lambda$.
Let $f$ be expressed using the half-range Fourier cosine series over $\openint 0 \lambda$:
:$\ds \map C x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} \lambda$
where:
:$a_n = \ds \frac 2 \lambda \int_0... | From Fourier Series for Even Function over Symmetric Range, $\map C x$ is the Fourier series of an even real function over the interval $\openint 0 \lambda$.
We have that $\map C x \sim \map f x$ over $\openint 0 \lambda$.
Thus over $\openint {-\lambda} 0$ it follows that $\map C x = \map f {-x}$.
{{qed}} | Let $\map f x$ be a [[Definition:Real Function|real function]] defined on the [[Definition:Open Real Interval|open real interval]] $\openint 0 \lambda$.
Let $f$ be expressed using the [[Definition:Half-Range Fourier Cosine Series|half-range Fourier cosine series]] over $\openint 0 \lambda$:
:$\ds \map C x \sim \frac ... | From [[Fourier Series for Even Function over Symmetric Range]], $\map C x$ is the [[Definition:Fourier Series|Fourier series]] of an [[Definition:Even Function|even]] [[Definition:Real Function|real function]] over the [[Definition:Closed Real Interval|interval]] $\openint 0 \lambda$.
We have that $\map C x \sim \map ... | Half-Range Fourier Cosine Series over Negative Range | https://proofwiki.org/wiki/Half-Range_Fourier_Cosine_Series_over_Negative_Range | https://proofwiki.org/wiki/Half-Range_Fourier_Cosine_Series_over_Negative_Range | [
"Half-Range Fourier Series"
] | [
"Definition:Real Function",
"Definition:Real Interval/Open",
"Definition:Half-Range Fourier Cosine Series",
"Definition:Real Interval/Closed",
"Definition:Real Function",
"Definition:Half-Range Fourier Cosine Series",
"Definition:Even Function"
] | [
"Fourier Series for Even Function over Symmetric Range",
"Definition:Fourier Series",
"Definition:Even Function",
"Definition:Real Function",
"Definition:Real Interval/Closed"
] |
proofwiki-14105 | Fourier Series/x over 0 to 2, x-2 over 2 to 4 | Let $\map f x$ be the real function defined on $\openint 0 4$ as:
$\quad \map f x = \begin{cases}
x & : 0 < x \le 2 \\
x - 2 & : 2 < x < 4 \end{cases}$
Then its Fourier series can be expressed as:
:$\ds \map f x \sim 1 + \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{r - 1} } {2 r - 1} \paren {1 + \frac {4... | Let $\map f x$ be the function defined as:
$\quad \forall x \in \openint 0 4: \begin{cases}
x & : 0 < x \le 2 \\
x - 2 & : 2 < x < 4 \end{cases}$
Let $f$ be expressed by a half-range Fourier cosine series:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos \frac {n \pi x} 4$
where for all $n \in \... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 4$ as:
$\quad \map f x = \begin{cases}
x & : 0 < x \le 2 \\
x - 2 & : 2 < x < 4 \end{cases}$
Then its [[Definition:Fourier Series|Fourier series]] can be expressed as:
:$\ds \map f x \sim 1 + \frac 4 \pi \sum_{n \mathop = 1}^\in... | Let $\map f x$ be the [[Definition:Real Function|function]] defined as:
$\quad \forall x \in \openint 0 4: \begin{cases}
x & : 0 < x \le 2 \\
x - 2 & : 2 < x < 4 \end{cases}$
Let $f$ be expressed by a [[Definition:Half-Range Fourier Cosine Series|half-range Fourier cosine series]]:
:$\ds \map f x \sim \frac {a_0} 2... | Fourier Series/x over 0 to 2, x-2 over 2 to 4 | https://proofwiki.org/wiki/Fourier_Series/x_over_0_to_2,_x-2_over_2_to_4 | https://proofwiki.org/wiki/Fourier_Series/x_over_0_to_2,_x-2_over_2_to_4 | [
"Examples of Fourier Series"
] | [
"Definition:Real Function",
"Definition:Fourier Series"
] | [
"Definition:Real Function",
"Definition:Half-Range Fourier Cosine Series",
"Primitive of Power",
"Linear Combination of Integrals/Definite",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Primitive of x by Cosine of a x",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multip... |
proofwiki-14106 | Fourier Series over General Range from Specific | Let $a, b \in \R$ be real numbers.
Let $f: \R \to \R$ be a function such that $\ds \int_a^b \map f x \rd x$ converges absolutely.
Then $f$ can be expressed by a Fourier series of the form:
:$\ds \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \p... | Consider the Fourier series:
:$(1): \quad \ds \map S x = \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \pi \paren {x - a} } {b - a} }$
Let $\xi = \dfrac {2 \pi \paren {x - a} } {b - a}$.
Then:
:$\dfrac {\d \xi} {\d x} = \dfrac {2 \pi} {b - a}$... | Let $a, b \in \R$ be [[Definition:Real Number|real numbers]].
Let $f: \R \to \R$ be a [[Definition:Function|function]] such that $\ds \int_a^b \map f x \rd x$ [[Definition:Absolute Convergence of Integral|converges absolutely]].
Then $f$ can be expressed by a [[Definition:Fourier Series|Fourier series]] of the form:... | Consider the [[Definition:Fourier Series|Fourier series]]:
:$(1): \quad \ds \map S x = \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \pi \paren {x - a} } {b - a} }$
Let $\xi = \dfrac {2 \pi \paren {x - a} } {b - a}$.
Then:
:$\dfrac {\d \xi}... | Fourier Series over General Range from Specific | https://proofwiki.org/wiki/Fourier_Series_over_General_Range_from_Specific | https://proofwiki.org/wiki/Fourier_Series_over_General_Range_from_Specific | [
"Fourier Series"
] | [
"Definition:Real Number",
"Definition:Function",
"Definition:Absolutely Convergent Integral",
"Definition:Fourier Series"
] | [
"Definition:Fourier Series",
"Definition:Fourier Series/Range 2 Pi",
"Definition:Real Function",
"Definition:Absolutely Convergent Integral",
"Definition:Fourier Series/Fourier Coefficient/Range 2 Pi",
"Definition:Fourier Series/Fourier Coefficient/Range 2 Pi",
"Definition:Fourier Series/Fourier Coeffic... |
proofwiki-14107 | Definite Integral to Infinity of Reciprocal of 1 plus Power of x | :$\ds \int_0^\infty \frac 1 {1 + x^n} \rd x = \frac \pi n \map \csc {\frac \pi n}$ | From Euler's Reflection Formula:
:$\map \Gamma {\dfrac 1 n} \map \Gamma {1 - \dfrac 1 n} = \pi \map \csc {\dfrac \pi n}$
Then:
{{begin-eqn}}
{{eqn | l = \map \Gamma {\frac 1 n} \map \Gamma {1 - \frac 1 n}
| r = \frac {\map \Gamma {\frac 1 n} \map \Gamma {1 - \frac 1 n} } {\map \Gamma {1 - \frac 1 n + \frac 1 n} }
| ... | :$\ds \int_0^\infty \frac 1 {1 + x^n} \rd x = \frac \pi n \map \csc {\frac \pi n}$ | From [[Euler's Reflection Formula]]:
:$\map \Gamma {\dfrac 1 n} \map \Gamma {1 - \dfrac 1 n} = \pi \map \csc {\dfrac \pi n}$
Then:
{{begin-eqn}}
{{eqn | l = \map \Gamma {\frac 1 n} \map \Gamma {1 - \frac 1 n}
| r = \frac {\map \Gamma {\frac 1 n} \map \Gamma {1 - \frac 1 n} } {\map \Gamma {1 - \frac 1 n + \frac 1 n... | Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_1_plus_Power_of_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_1_plus_Power_of_x/Proof_1 | [
"Definite Integral to Infinity of Reciprocal of 1 plus Power of x",
"Definite Integrals",
"Examples of Definite Integrals"
] | [] | [
"Euler's Reflection Formula",
"Derivative of Composite Function",
"Derivative of Tangent Function",
"Power Rule for Derivatives",
"Sum of Squares of Sine and Cosine/Corollary 1",
"Integration by Substitution"
] |
proofwiki-14108 | Definite Integral to Infinity of Reciprocal of 1 plus Power of x | :$\ds \int_0^\infty \frac 1 {1 + x^n} \rd x = \frac \pi n \map \csc {\frac \pi n}$ | Let $R > 1$ be a real number.
Let:
:$\ds C_R = \set {R e^{i \theta} : 0 \le \theta \le \frac {2 \pi} n}$
{{explain|the fact that we are working in the complex plane. This is not immediately obvious.}}
Let $L_R$ be the straight line segment from $0$ to $R e^{\frac {2 \pi i} n}$.
Let $\Gamma_R = \closedint 0 R \cup C_R \... | :$\ds \int_0^\infty \frac 1 {1 + x^n} \rd x = \frac \pi n \map \csc {\frac \pi n}$ | Let $R > 1$ be a [[Definition:Real Number|real number]].
Let:
:$\ds C_R = \set {R e^{i \theta} : 0 \le \theta \le \frac {2 \pi} n}$
{{explain|the fact that we are working in the complex plane. This is not immediately obvious.}}
Let $L_R$ be the [[Definition:Straight Line Segment|straight line segment]] from $0$ to ... | Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_1_plus_Power_of_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_1_plus_Power_of_x/Proof_2 | [
"Definite Integral to Infinity of Reciprocal of 1 plus Power of x",
"Definite Integrals",
"Examples of Definite Integrals"
] | [] | [
"Definition:Real Number",
"Definition:Line/Straight Line Segment",
"Definition:Anticlockwise",
"Estimation Lemma for Contour Integrals",
"Definition:Integration/Integrand",
"Definition:Meromorphic Function",
"Definition:Order of Pole/Simple Pole",
"Definition:Contour/Closed/Complex Plane",
"Cauchy's... |
proofwiki-14109 | Definite Integral to Infinity of Reciprocal of 1 plus Power of x | :$\ds \int_0^\infty \frac 1 {1 + x^n} \rd x = \frac \pi n \map \csc {\frac \pi n}$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = \paren {\tan \theta}^{\frac 2 n}
| c = Integration by Substitution
}}
{{eqn | ll= \leadsto
| l = \rd x
| r = \frac 2 n \paren {\tan \theta}^{\frac 2 n - 1} \sec^2 \theta \rd \theta
| c = Derivative of Tangent Function, Derivative of Power, Chain Rule... | :$\ds \int_0^\infty \frac 1 {1 + x^n} \rd x = \frac \pi n \map \csc {\frac \pi n}$ | Let:
{{begin-eqn}}
{{eqn | l = x
| r = \paren {\tan \theta}^{\frac 2 n}
| c = [[Integration by Substitution]]
}}
{{eqn | ll= \leadsto
| l = \rd x
| r = \frac 2 n \paren {\tan \theta}^{\frac 2 n - 1} \sec^2 \theta \rd \theta
| c = [[Derivative of Tangent Function]], [[Derivative of Power]]... | Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Proof 3 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_1_plus_Power_of_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_1_plus_Power_of_x/Proof_3 | [
"Definite Integral to Infinity of Reciprocal of 1 plus Power of x",
"Definite Integrals",
"Examples of Definite Integrals"
] | [] | [
"Integration by Substitution",
"Derivative of Tangent Function",
"Power Rule for Derivatives",
"Derivative of Composite Function",
"Tangent of Zero",
"Tangent of Right Angle",
"Integration by Substitution",
"Sum of Squares of Sine and Cosine",
"Euler's Reflection Formula"
] |
proofwiki-14110 | Leibniz's Integral Rule | Let $\map f {x, t}$, $\map a t$, $\map b t$ be continuously differentiable real functions on some region $R$ of the $\tuple {x, t}$ plane.
Then for all $\tuple {x, t} \in R$:
:$\ds \frac \d {\d t} \int_{\map a t}^{\map b t} \map f {x, t} \rd x = \map f {\map b t, t} \frac {\d b} {\d t} - \map f {\map a t, t} \frac {\d ... | {{begin-eqn}}
{{eqn | l = \frac \d {\d t} \int_{\map a t}^{\map b t} \map f {x, t} \rd x
| r = \lim_{h \mathop \to 0} \frac 1 h \paren {\int_{\map a {t + h} }^{\map b {t + h} } \map f {x, t + h} \rd x - \int_{\map a t}^{\map b t} \map f {x, t} \rd x }
| c = {{Defof|Derivative of Real Function}}
}}
{{eqn | r... | Let $\map f {x, t}$, $\map a t$, $\map b t$ be [[Definition:Continuously Differentiable Real-Valued Function|continuously differentiable]] [[Definition:Real Function|real functions]] on some region $R$ of the $\tuple {x, t}$ [[Definition:Cartesian Plane|plane]].
Then for all $\tuple {x, t} \in R$:
:$\ds \frac \d {\d... | {{begin-eqn}}
{{eqn | l = \frac \d {\d t} \int_{\map a t}^{\map b t} \map f {x, t} \rd x
| r = \lim_{h \mathop \to 0} \frac 1 h \paren {\int_{\map a {t + h} }^{\map b {t + h} } \map f {x, t + h} \rd x - \int_{\map a t}^{\map b t} \map f {x, t} \rd x }
| c = {{Defof|Derivative of Real Function}}
}}
{{eqn | r... | Leibniz's Integral Rule | https://proofwiki.org/wiki/Leibniz's_Integral_Rule | https://proofwiki.org/wiki/Leibniz's_Integral_Rule | [
"Leibniz's Integral Rule",
"Definite Integrals"
] | [
"Definition:Continuously Differentiable/Real-Valued Function",
"Definition:Real Function",
"Definition:Cartesian Plane"
] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Linear Combination of Integrals",
"Mean Value Theorem for Integrals",
"Combination Theorem for Limits of Functions/Real",
"Linear Combination of Integrals",
"Limit of Function by Convergent Sequences",
"Extreme Value Theorem",
"Mean V... |
proofwiki-14111 | Series Expansion of Function over Complete Orthonormal Set | Let $\map f x$ be a real function defined over the interval $\openint a b$.
Let $\map f x$ be able to be expressed in terms of a complete orthonormal set of real functions $S := \family {\map {\phi_i} x}_{i \mathop \in I}$ for some indexing set $I$:
:$\map f x = \ds \sum_{i \mathop \in I} a_i \map {\phi_i} x$
Then the ... | {{begin-eqn}}
{{eqn | l = \map f x
| r = \sum_{i \mathop \in I} a_i \map {\phi_i} x
| c =
}}
{{eqn | ll= \leadsto
| l = \map f x \map {\phi_n} x
| r = \sum_{i \mathop \in I} a_i \map {\phi_i} x \map {\phi_n} x
| c = for arbitrary $n \in I$
}}
{{eqn | ll= \leadsto
| l = \int_a^b \map... | Let $\map f x$ be a [[Definition:Real Function|real function]] defined over the [[Definition:Real Interval|interval]] $\openint a b$.
Let $\map f x$ be able to be expressed in terms of a [[Definition:Complete Orthonormal Set of Real Functions|complete orthonormal set of real functions]] $S := \family {\map {\phi_i} x}... | {{begin-eqn}}
{{eqn | l = \map f x
| r = \sum_{i \mathop \in I} a_i \map {\phi_i} x
| c =
}}
{{eqn | ll= \leadsto
| l = \map f x \map {\phi_n} x
| r = \sum_{i \mathop \in I} a_i \map {\phi_i} x \map {\phi_n} x
| c = for arbitrary $n \in I$
}}
{{eqn | ll= \leadsto
| l = \int_a^b \map... | Series Expansion of Function over Complete Orthonormal Set | https://proofwiki.org/wiki/Series_Expansion_of_Function_over_Complete_Orthonormal_Set | https://proofwiki.org/wiki/Series_Expansion_of_Function_over_Complete_Orthonormal_Set | [
"Orthonormal Sets"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Complete Orthonormal Set of Real Functions",
"Definition:Indexing Set"
] | [] |
proofwiki-14112 | Vector as Sum of Orthogonal Base Vectors | Let $\mathbf v$ be a vector quantity in ordinary $3$-space.
Let $\mathbf i, \mathbf j, \mathbf k$ be orthonormal base vectors.
Then:
:$\mathbf v = \paren {\mathbf v \cdot \mathbf i} \mathbf i + \paren {\mathbf v \cdot \mathbf j} \mathbf j + \paren {\mathbf v \cdot \mathbf k} \mathbf k$ | {{ProofWanted|Background needed}} | Let $\mathbf v$ be a [[Definition:Vector Quantity|vector quantity]] in [[Definition:Ordinary Space|ordinary $3$-space]].
Let $\mathbf i, \mathbf j, \mathbf k$ be [[Definition:Orthonormal Base Vector|orthonormal base vectors]].
Then:
:$\mathbf v = \paren {\mathbf v \cdot \mathbf i} \mathbf i + \paren {\mathbf v \cdot ... | {{ProofWanted|Background needed}} | Vector as Sum of Orthogonal Base Vectors | https://proofwiki.org/wiki/Vector_as_Sum_of_Orthogonal_Base_Vectors | https://proofwiki.org/wiki/Vector_as_Sum_of_Orthogonal_Base_Vectors | [
"Orthonormal Sets",
"Vector Algebra"
] | [
"Definition:Vector Quantity",
"Definition:Ordinary Space",
"Definition:Orthonormal Base Vector"
] | [] |
proofwiki-14113 | Scaled Sine Functions of Integer Multiples form Orthonormal Set | For all $n \in \Z_{>0}$, let $\map {\phi_n} x$ be the real function defined on the interval $\openint 0 \lambda$ as:
:$\map {\phi_n} x = \sqrt {\dfrac 2 \lambda} \sin \dfrac {n \pi x} \lambda$
Let $S$ be the set:
:$S = \set {\phi_n: n \in \Z_{>0} }$
Then $S$ is an orthonormal set. | Consider the definite integral:
:$I_{m n} = \ds \int_0^\lambda \map {\phi_m} x \map {\phi_n} x \rd x$
From Sine Function is Odd, each of $\map {\phi_n} x$ is an odd function.
From Odd Function Times Odd Function is Even, $\map {\phi_m} x \map {\phi_n} x$ is even.
That is:
:$\paren {\sqrt {\dfrac 2 \lambda} \sin \dfrac ... | For all $n \in \Z_{>0}$, let $\map {\phi_n} x$ be the [[Definition:Real Function|real function]] defined on the [[Definition:Real Interval|interval]] $\openint 0 \lambda$ as:
:$\map {\phi_n} x = \sqrt {\dfrac 2 \lambda} \sin \dfrac {n \pi x} \lambda$
Let $S$ be the [[Definition:Set|set]]:
:$S = \set {\phi_n: n \in \Z_... | Consider the [[Definition:Definite Integral|definite integral]]:
:$I_{m n} = \ds \int_0^\lambda \map {\phi_m} x \map {\phi_n} x \rd x$
From [[Sine Function is Odd]], each of $\map {\phi_n} x$ is an [[Definition:Odd Function|odd function]].
From [[Odd Function Times Odd Function is Even]], $\map {\phi_m} x \map {\phi... | Scaled Sine Functions of Integer Multiples form Orthonormal Set | https://proofwiki.org/wiki/Scaled_Sine_Functions_of_Integer_Multiples_form_Orthonormal_Set | https://proofwiki.org/wiki/Scaled_Sine_Functions_of_Integer_Multiples_form_Orthonormal_Set | [
"Orthonormal Sets",
"Sine Function"
] | [
"Definition:Real Function",
"Definition:Real Interval",
"Definition:Set",
"Definition:Orthonormal Set of Real Functions"
] | [
"Definition:Definite Integral",
"Sine Function is Odd",
"Definition:Odd Function",
"Odd Function Times Odd Function is Even",
"Definition:Even Function",
"Definition:Even Function",
"Linear Combination of Integrals/Definite",
"Integration by Substitution",
"Linear Combination of Integrals/Definite",... |
proofwiki-14114 | Correspondence Theorem for Quotient Rings/Bijection | The direct image mapping $\pi^\to$ and the inverse image mapping $\pi^\gets$ induce reverse bijections between the ideals of $A$ containing $\mathfrak a$ and the ideals of $A/\mathfrak a$, specifically:
Let $I$ be the set of ideals of $A$ containing $\mathfrak a$.
Let $J$ be the set of ideals of $A / \mathfrak a$.
Then... | Follows from Correspondence Theorem for Ring Epimorphisms/Bijection
{{qed}}
Category:Quotient Rings
0snouo05fh0g4n0ggttzc3fcfc09bmb | The [[Definition:Direct Image Mapping|direct image mapping]] $\pi^\to$ and the [[Definition:Inverse Image Mapping|inverse image mapping]] $\pi^\gets$ induce [[Definition:Reverse Bijections|reverse bijections]] between the [[Definition:Ideal of Ring|ideals]] of $A$ [[Definition:Set Containment|containing]] $\mathfrak a$... | Follows from [[Correspondence Theorem for Ring Epimorphisms/Bijection]]
{{qed}}
[[Category:Quotient Rings]]
0snouo05fh0g4n0ggttzc3fcfc09bmb | Correspondence Theorem for Quotient Rings/Bijection | https://proofwiki.org/wiki/Correspondence_Theorem_for_Quotient_Rings/Bijection | https://proofwiki.org/wiki/Correspondence_Theorem_for_Quotient_Rings/Bijection | [
"Quotient Rings"
] | [
"Definition:Direct Image Mapping",
"Definition:Inverse Image Mapping",
"Definition:Inverse Mapping",
"Definition:Ideal of Ring",
"Definition:Subset",
"Definition:Set",
"Definition:Ideal of Ring",
"Definition:Subset",
"Definition:Set",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring",
"De... | [
"Correspondence Theorem for Ring Epimorphisms/Bijection",
"Category:Quotient Rings"
] |
proofwiki-14115 | Correspondence Theorem for Ring Epimorphisms/Bijection | The direct image mapping $\pi^\to$ and the inverse image mapping $\pi^\gets$ induce reverse bijections between $I$ and $J$, specifically:
#For every ideal $\mathfrak a \in I$, its image $\pi^{\to}(\mathfrak a) = \pi(\mathfrak a) \in J$.
#For every ideal $\mathfrak b \in J$, its preimage $\pi^{\gets}(\mathfrak b) = \pi^... | The first statements follow from:
* Preimage of Ideal under Ring Homomorphism is Ideal
* Image of Ideal under Ring Epimorphism is Ideal
The last statement follows from:
* Image of Preimage of Ideal under Ring Epimorphism
* Preimage of Image of Ideal under Ring Homomorphism
{{qed}}
Category:Ring Homomorphisms
6brohngslt... | The [[Definition:Direct Image Mapping|direct image mapping]] $\pi^\to$ and the [[Definition:Inverse Image Mapping|inverse image mapping]] $\pi^\gets$ induce [[Definition:Reverse Bijections|reverse bijections]] between $I$ and $J$, specifically:
#For every [[Definition:Ideal of Ring|ideal]] $\mathfrak a \in I$, its [[D... | The first statements follow from:
* [[Preimage of Ideal under Ring Homomorphism is Ideal]]
* [[Image of Ideal under Ring Epimorphism is Ideal]]
The last statement follows from:
* [[Image of Preimage of Ideal under Ring Epimorphism]]
* [[Preimage of Image of Ideal under Ring Homomorphism]]
{{qed}}
[[Category:Ring Homo... | Correspondence Theorem for Ring Epimorphisms/Bijection | https://proofwiki.org/wiki/Correspondence_Theorem_for_Ring_Epimorphisms/Bijection | https://proofwiki.org/wiki/Correspondence_Theorem_for_Ring_Epimorphisms/Bijection | [
"Ring Homomorphisms"
] | [
"Definition:Direct Image Mapping",
"Definition:Inverse Image Mapping",
"Definition:Inverse Mapping",
"Definition:Ideal of Ring",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Ideal of Ring",
"Definition:Preimage/Mapping/Subset",
"Definition:Restriction/Mapping",
"Definition:Inverse Map... | [
"Preimage of Ideal under Ring Homomorphism is Ideal",
"Image of Ideal under Ring Epimorphism is Ideal",
"Image of Preimage of Ideal under Ring Epimorphism",
"Preimage of Image of Ideal under Ring Homomorphism",
"Category:Ring Homomorphisms"
] |
proofwiki-14116 | Image of Ideal under Ring Epimorphism is Ideal | Let $f : A \to B$ be a ring epimorphism.
Let $I \subseteq A$ be an ideal.
Then its image $f(I) \subseteq B$ is an ideal. | {{proof wanted}}
Category:Ring Homomorphisms
k1fwp02uwn3fgkog4t5wmwpm3xykfze | Let $f : A \to B$ be a [[Definition:Ring Epimorphism|ring epimorphism]].
Let $I \subseteq A$ be an [[Definition:Ideal of Ring|ideal]].
Then its [[Definition:Image of Subset under Mapping|image]] $f(I) \subseteq B$ is an [[Definition:Ideal of Ring|ideal]]. | {{proof wanted}}
[[Category:Ring Homomorphisms]]
k1fwp02uwn3fgkog4t5wmwpm3xykfze | Image of Ideal under Ring Epimorphism is Ideal | https://proofwiki.org/wiki/Image_of_Ideal_under_Ring_Epimorphism_is_Ideal | https://proofwiki.org/wiki/Image_of_Ideal_under_Ring_Epimorphism_is_Ideal | [
"Ring Homomorphisms"
] | [
"Definition:Ring Epimorphism",
"Definition:Ideal of Ring",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Ideal of Ring"
] | [
"Category:Ring Homomorphisms"
] |
proofwiki-14117 | Preimage of Prime Ideal under Ring Homomorphism is Prime Ideal | Let $A$ and $B$ be commutative rings with unity.
Let $f : A \to B$ be a ring homomorphism.
Let $\mathfrak p \subseteq B$ be a prime ideal.
Then its preimage $\map {f^{-1} } {\mathfrak p}$ is a prime ideal of $A$. | By Preimage of Ideal under Ring Homomorphism is Ideal, $\map {f^{-1} } {\mathfrak p}$ is a ideal of $A$.
Let $B / \mathfrak p$ be the quotient ring.
By Prime Ideal iff Quotient Ring is Integral Domain, $B / \mathfrak p$ is an integral domain.
By Quotient Ring of Kernel of Ring Epimorphism, the quotient ring epimorphism... | Let $A$ and $B$ be [[Definition:Commutative Ring with Unity|commutative rings with unity]].
Let $f : A \to B$ be a [[Definition:Unital Ring Homomorphism|ring homomorphism]].
Let $\mathfrak p \subseteq B$ be a [[Definition:Prime Ideal of Ring|prime ideal]].
Then its [[Definition:Preimage of Subset under Mapping|prei... | By [[Preimage of Ideal under Ring Homomorphism is Ideal]], $\map {f^{-1} } {\mathfrak p}$ is a [[Definition:Ideal of Ring|ideal]] of $A$.
Let $B / \mathfrak p$ be the [[Definition:Quotient Ring|quotient ring]].
By [[Prime Ideal iff Quotient Ring is Integral Domain]], $B / \mathfrak p$ is an [[Definition:Integral Doma... | Preimage of Prime Ideal under Ring Homomorphism is Prime Ideal/Proof 2 | https://proofwiki.org/wiki/Preimage_of_Prime_Ideal_under_Ring_Homomorphism_is_Prime_Ideal | https://proofwiki.org/wiki/Preimage_of_Prime_Ideal_under_Ring_Homomorphism_is_Prime_Ideal/Proof_2 | [
"Prime Ideals of Rings",
"Preimage of Prime Ideal under Ring Homomorphism is Prime Ideal"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Unital Ring Homomorphism",
"Definition:Prime Ideal of Ring",
"Definition:Preimage/Mapping/Subset",
"Definition:Prime Ideal of Ring"
] | [
"Preimage of Ideal under Ring Homomorphism is Ideal",
"Definition:Ideal of Ring",
"Definition:Quotient Ring",
"Prime Ideal iff Quotient Ring is Integral Domain",
"Definition:Integral Domain",
"Quotient Ring of Kernel of Ring Epimorphism",
"Definition:Quotient Epimorphism/Ring",
"Definition:Kernel of R... |
proofwiki-14118 | Integral Domain iff Zero Ideal is Prime | Let $A$ be a commutative ring with unity.
{{TFAE}}
:$(1): \quad A$ is an integral domain
:$(2): \quad$ the zero ideal $\ideal 0 \subseteq A$ is prime | By Prime Ideal iff Quotient Ring is Integral Domain, $\ideal 0$ is prime {{iff}} the quotient ring $A / \ideal 0$ is an integral domain.
By Quotient Ring by Null Ideal, $A \cong A / \ideal 0$.
{{qed}}
Category:Integral Domains
Category:Prime Ideals of Rings
47fvl892omw913l26myb2x9f28eq91b | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
{{TFAE}}
:$(1): \quad A$ is an [[Definition:Integral Domain|integral domain]]
:$(2): \quad$ the [[Definition:Zero Ideal|zero ideal]] $\ideal 0 \subseteq A$ is [[Definition:Prime Ideal of Ring|prime]] | By [[Prime Ideal iff Quotient Ring is Integral Domain]], $\ideal 0$ is [[Definition:Prime Ideal of Ring|prime]] {{iff}} the [[Definition:Quotient Ring|quotient ring]] $A / \ideal 0$ is an [[Definition:Integral Domain|integral domain]].
By [[Quotient Ring by Null Ideal]], $A \cong A / \ideal 0$.
{{qed}}
[[Category:Int... | Integral Domain iff Zero Ideal is Prime | https://proofwiki.org/wiki/Integral_Domain_iff_Zero_Ideal_is_Prime | https://proofwiki.org/wiki/Integral_Domain_iff_Zero_Ideal_is_Prime | [
"Integral Domains",
"Prime Ideals of Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Integral Domain",
"Definition:Null Ideal",
"Definition:Prime Ideal of Ring"
] | [
"Prime Ideal iff Quotient Ring is Integral Domain",
"Definition:Prime Ideal of Ring",
"Definition:Quotient Ring",
"Definition:Integral Domain",
"Quotient Ring by Null Ideal",
"Category:Integral Domains",
"Category:Prime Ideals of Rings"
] |
proofwiki-14119 | Kernel of Composition of Ring Homomorphisms | Let $f : A \to B$ and $g : B \to C$ be ring homomorphisms.
Let $g \circ f$ be their composition.
Then the kernel of $g \circ f$ is the preimage under $f$ of the kernel of $g$:
:$\map \ker {g \circ f} = f^{-1} \sqbrk {\ker g}$ | By definition, the kernel of a ring homomorphism is its kernel when considered as a group homomorphism between the additive groups.
The result follows from Kernel of Composition of Group Homomorphisms.
{{qed}}
Category:Ring Homomorphisms
sdjqic2fcbk30acpe13xsgurfciujpe | Let $f : A \to B$ and $g : B \to C$ be [[Definition:Ring Homomorphism|ring homomorphisms]].
Let $g \circ f$ be their [[Definition:Composition of Mappings|composition]].
Then the [[Definition:Kernel of Ring Homomorphism|kernel]] of $g \circ f$ is the [[Definition:Preimage of Subset under Mapping|preimage]] under $f$ ... | By definition, the [[Definition:Kernel of Ring Homomorphism|kernel]] of a [[Definition:Ring Homomorphism|ring homomorphism]] is its [[Definition:Kernel of Group Homomorphism|kernel]] when considered as a [[Definition:Group Homomorphism|group homomorphism]] between the [[definition:Additive Group of Ring|additive groups... | Kernel of Composition of Ring Homomorphisms | https://proofwiki.org/wiki/Kernel_of_Composition_of_Ring_Homomorphisms | https://proofwiki.org/wiki/Kernel_of_Composition_of_Ring_Homomorphisms | [
"Ring Homomorphisms"
] | [
"Definition:Ring Homomorphism",
"Definition:Composition of Mappings",
"Definition:Kernel of Ring Homomorphism",
"Definition:Preimage/Mapping/Subset",
"Definition:Kernel of Ring Homomorphism"
] | [
"Definition:Kernel of Ring Homomorphism",
"Definition:Ring Homomorphism",
"Definition:Kernel of Group Homomorphism",
"Definition:Group Homomorphism",
"definition:Additive Group of Ring",
"Kernel of Composition of Group Homomorphisms",
"Category:Ring Homomorphisms"
] |
proofwiki-14120 | Ring Epimorphism Preserves Unity | Let $A$ be a ring with unity $1$.
Let $B$ be a ring.
Let $f: A \to B$ be a ring epimorphism.
Then $\map f 1$ is a unity of $B$. | By definition, $f$ is a semigroup homomorphism between multiplicative semigroups.
A unity of a ring is by definition an identity element of its multiplicative semigroup.
Thus the result follows from Epimorphism Preserves Identity.
{{qed}}
Category:Ring Epimorphisms
9nofbcnttzpsssgf79g3ug472j6fwcu | Let $A$ be a [[Definition:Ring with Unity|ring with unity]] $1$.
Let $B$ be a [[Definition:Ring (Abstract Algebra)|ring]].
Let $f: A \to B$ be a [[Definition:Ring Epimorphism|ring epimorphism]].
Then $\map f 1$ is a [[Definition:Unity of Ring|unity]] of $B$. | By definition, $f$ is a [[Definition:Semigroup Homomorphism|semigroup homomorphism]] between [[Definition:Multiplicative Semigroup of Ring|multiplicative semigroups]].
A [[Definition:Unity of Ring|unity]] of a [[Definition:Ring (Abstract Algebra)|ring]] is by definition an [[Definition:Identity Element|identity elemen... | Ring Epimorphism Preserves Unity | https://proofwiki.org/wiki/Ring_Epimorphism_Preserves_Unity | https://proofwiki.org/wiki/Ring_Epimorphism_Preserves_Unity | [
"Ring Epimorphisms"
] | [
"Definition:Ring with Unity",
"Definition:Ring (Abstract Algebra)",
"Definition:Ring Epimorphism",
"Definition:Unity (Abstract Algebra)/Ring"
] | [
"Definition:Semigroup Homomorphism",
"Definition:Multiplicative Semigroup of Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Ring (Abstract Algebra)",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Multiplicative Semigroup of Ring",
"Epimorphism Preserves Identit... |
proofwiki-14121 | Complement of Prime Ideal of Ring is Multiplicatively Closed | Let $R$ be a commutative ring with unity.
Let $P \subset R$ be a prime ideal of $R$.
Then its complement $R \setminus P$ is multiplicatively closed. | Since $P$ is a proper ideal by {{Defof|Prime Ideal of Commutative and Unitary Ring|Prime Ideal}}, we have:
:$1_R \in R \setminus P$
where $1_R$ is the unity of $R$.
{{AimForCont}} $R \setminus P$ is not multiplicatively closed.
That is:
:$\exists a, b \in R \setminus P: a b \notin R \setminus P$
This means:
:$a b \in P... | Let $R$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $P \subset R$ be a [[Definition:Prime Ideal of Commutative and Unitary Ring|prime ideal]] of $R$.
Then its [[Definition:Relative Complement|complement]] $R \setminus P$ is [[Definition:Multiplicatively Closed Subset of Ring|mul... | Since $P$ is a [[Definition:Proper Ideal of Ring|proper ideal]] by {{Defof|Prime Ideal of Commutative and Unitary Ring|Prime Ideal}}, we have:
:$1_R \in R \setminus P$
where $1_R$ is the [[Definition:Unity of Ring|unity]] of $R$.
{{AimForCont}} $R \setminus P$ is not [[Definition:Multiplicatively Closed Subset of Rin... | Complement of Prime Ideal of Ring is Multiplicatively Closed | https://proofwiki.org/wiki/Complement_of_Prime_Ideal_of_Ring_is_Multiplicatively_Closed | https://proofwiki.org/wiki/Complement_of_Prime_Ideal_of_Ring_is_Multiplicatively_Closed | [
"Prime Ideals of Rings",
"Localization of Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Prime Ideal of Ring/Commutative and Unitary Ring",
"Definition:Relative Complement",
"Definition:Multiplicatively Closed Subset of Ring"
] | [
"Definition:Ideal of Ring/Proper Ideal",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Multiplicatively Closed Subset of Ring",
"Definition:Contradiction",
"Definition:Prime Ideal of Ring/Commutative and Unitary Ring"
] |
proofwiki-14122 | Derivative of Gamma Function | :$\ds \map {\Gamma'} x = \int_0^\infty t^{x - 1} \ln t \, e^{-t} \rd t$
where $\map {\Gamma'} x$ denotes the derivative of the Gamma function evaluated at $x$. | {{begin-eqn}}
{{eqn | l = \map {\Gamma'} x
| r = \frac \d {\d x} \int_0^\infty t^{x - 1} e^{-t} \rd t
| c = {{Defof|Gamma Function}}
}}
{{eqn | r = \int_0^\infty \frac {\partial} {\partial x} t^{x - 1} e^{-t} \rd t
| c = Leibniz's Integral Rule
}}
{{eqn | r = \int_0^\infty t^{x-1} \ln t \, e^{-t} \rd t
| c = Deriva... | :$\ds \map {\Gamma'} x = \int_0^\infty t^{x - 1} \ln t \, e^{-t} \rd t$
where $\map {\Gamma'} x$ denotes the derivative of the [[Definition:Gamma Function|Gamma function]] evaluated at $x$. | {{begin-eqn}}
{{eqn | l = \map {\Gamma'} x
| r = \frac \d {\d x} \int_0^\infty t^{x - 1} e^{-t} \rd t
| c = {{Defof|Gamma Function}}
}}
{{eqn | r = \int_0^\infty \frac {\partial} {\partial x} t^{x - 1} e^{-t} \rd t
| c = [[Leibniz's Integral Rule]]
}}
{{eqn | r = \int_0^\infty t^{x-1} \ln t \, e^{-t} \rd t
| c = [[... | Derivative of Gamma Function | https://proofwiki.org/wiki/Derivative_of_Gamma_Function | https://proofwiki.org/wiki/Derivative_of_Gamma_Function | [
"Gamma Function",
"Derivatives"
] | [
"Definition:Gamma Function"
] | [
"Leibniz's Integral Rule",
"Derivative of General Exponential Function",
"Category:Gamma Function",
"Category:Derivatives"
] |
proofwiki-14123 | Derivative of Gamma Function | :$\ds \map {\Gamma'} x = \int_0^\infty t^{x - 1} \ln t \, e^{-t} \rd t$
where $\map {\Gamma'} x$ denotes the derivative of the Gamma function evaluated at $x$. | From Reciprocal times Derivative of Gamma Function:
:$\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z} = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }$
Setting $z = 1$:
{{begin-eqn}}
{{eqn | l = \frac {\map {\Gamma'} 1} {\map \Gamma 1}
| r = -\gamma + \sum_{n \mathop = 1}^\infty \pa... | :$\ds \map {\Gamma'} x = \int_0^\infty t^{x - 1} \ln t \, e^{-t} \rd t$
where $\map {\Gamma'} x$ denotes the derivative of the [[Definition:Gamma Function|Gamma function]] evaluated at $x$. | From [[Reciprocal times Derivative of Gamma Function]]:
:$\ds \dfrac {\map {\Gamma'} z} {\map \Gamma z} = -\gamma + \sum_{n \mathop = 1}^\infty \paren {\frac 1 n - \frac 1 {z + n - 1} }$
Setting $z = 1$:
{{begin-eqn}}
{{eqn | l = \frac {\map {\Gamma'} 1} {\map \Gamma 1}
| r = -\gamma + \sum_{n \mathop = 1}^\in... | Derivative of Gamma Function at 1/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Gamma_Function | https://proofwiki.org/wiki/Derivative_of_Gamma_Function_at_1/Proof_1 | [
"Gamma Function",
"Derivatives"
] | [
"Definition:Gamma Function"
] | [
"Reciprocal times Derivative of Gamma Function",
"Gamma Function Extends Factorial"
] |
proofwiki-14124 | Derivative of Gamma Function | :$\ds \map {\Gamma'} x = \int_0^\infty t^{x - 1} \ln t \, e^{-t} \rd t$
where $\map {\Gamma'} x$ denotes the derivative of the Gamma function evaluated at $x$. | {{begin-eqn}}
{{eqn | l = \frac 1 {\map \Gamma z}
| r = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }
| c = Weierstrass Form of Gamma Function
}}
{{eqn | ll= \leadsto
| l = \map \Gamma z
| r = \frac {e^{-\gamma z} } z \prod_{n \mathop = 1}^\infty \frac {... | :$\ds \map {\Gamma'} x = \int_0^\infty t^{x - 1} \ln t \, e^{-t} \rd t$
where $\map {\Gamma'} x$ denotes the derivative of the [[Definition:Gamma Function|Gamma function]] evaluated at $x$. | {{begin-eqn}}
{{eqn | l = \frac 1 {\map \Gamma z}
| r = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }
| c = [[Definition:Weierstrass Form of Gamma Function|Weierstrass Form of Gamma Function]]
}}
{{eqn | ll= \leadsto
| l = \map \Gamma z
| r = \frac {e^{-... | Reciprocal times Derivative of Gamma Function/Proof 1 | https://proofwiki.org/wiki/Derivative_of_Gamma_Function | https://proofwiki.org/wiki/Reciprocal_times_Derivative_of_Gamma_Function/Proof_1 | [
"Gamma Function",
"Derivatives"
] | [
"Definition:Gamma Function"
] | [
"Definition:Gamma Function/Weierstrass Form",
"Definition:Reciprocal",
"Definition:Differentiation",
"Product Rule for Derivatives",
"Definition:Continued Product",
"Definition:Division/Field/Complex Numbers"
] |
proofwiki-14125 | Derivative of Gamma Function | :$\ds \map {\Gamma'} x = \int_0^\infty t^{x - 1} \ln t \, e^{-t} \rd t$
where $\map {\Gamma'} x$ denotes the derivative of the Gamma function evaluated at $x$. | {{begin-eqn}}
{{eqn | l = \frac 1 {\map \Gamma z}
| r = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }
| c = Weierstrass Form of Gamma Function
}}
{{eqn | ll= \leadsto
| l = \map \Gamma z
| r = \frac {e^{-\gamma z} } z \prod_{n \mathop = 1}^\infty \frac {... | :$\ds \map {\Gamma'} x = \int_0^\infty t^{x - 1} \ln t \, e^{-t} \rd t$
where $\map {\Gamma'} x$ denotes the derivative of the [[Definition:Gamma Function|Gamma function]] evaluated at $x$. | {{begin-eqn}}
{{eqn | l = \frac 1 {\map \Gamma z}
| r = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }
| c = [[Definition:Weierstrass Form of Gamma Function|Weierstrass Form of Gamma Function]]
}}
{{eqn | ll= \leadsto
| l = \map \Gamma z
| r = \frac {e^{-... | Reciprocal times Derivative of Gamma Function/Proof 2 | https://proofwiki.org/wiki/Derivative_of_Gamma_Function | https://proofwiki.org/wiki/Reciprocal_times_Derivative_of_Gamma_Function/Proof_2 | [
"Gamma Function",
"Derivatives"
] | [
"Definition:Gamma Function"
] | [
"Definition:Gamma Function/Weierstrass Form",
"Definition:Reciprocal",
"Definition:Natural Logarithm",
"Sum of Logarithms",
"Difference of Logarithms",
"Logarithm of Power",
"Natural Logarithm of e is 1",
"Definition:Differentiation",
"Derivative of Composite Function",
"Derivative of Natural Loga... |
proofwiki-14126 | Hankel Representation of Riemann Zeta Function | Let $C$ be the Hankel contour.
Then for $s \in \C \setminus \Z_{>0}$:
:$\ds \map \zeta s = \frac {i \Gamma \paren {1 - s} } {2 \pi} \oint_C \frac {\paren {-z}^{s - 1} } {e^z - 1} \rd z$
where:
:$\zeta$ is the Riemann Zeta function
:$\Gamma$ is the Gamma function. | {{ProofWanted}}
{{Namedfor|Hermann Hankel|cat = Hankel}} | Let $C$ be the [[Definition:Hankel Contour|Hankel contour]].
Then for $s \in \C \setminus \Z_{>0}$:
:$\ds \map \zeta s = \frac {i \Gamma \paren {1 - s} } {2 \pi} \oint_C \frac {\paren {-z}^{s - 1} } {e^z - 1} \rd z$
where:
:$\zeta$ is the [[Definition:Riemann Zeta Function|Riemann Zeta function]]
:$\Gamma$ is the [... | {{ProofWanted}}
{{Namedfor|Hermann Hankel|cat = Hankel}} | Hankel Representation of Riemann Zeta Function | https://proofwiki.org/wiki/Hankel_Representation_of_Riemann_Zeta_Function | https://proofwiki.org/wiki/Hankel_Representation_of_Riemann_Zeta_Function | [
"Riemann Zeta Function"
] | [
"Definition:Hankel Contour",
"Definition:Riemann Zeta Function",
"Definition:Gamma Function"
] | [] |
proofwiki-14127 | Fourier Series/Minus Pi over 0 to Pi, x minus Pi over Pi to 2 Pi | Let $\map f x$ be the real function defined on $\openint 0 {2 \pi}$ as:
:500pxthumbright$\map f x$ and its $7$th approximation
:<nowiki>$\map f x = \begin{cases}
-\pi & : 0 < x \le \pi \\
x - \pi & : \pi < x < 2 \pi \end{cases}$</nowiki>
Then its Fourier series can be expressed as:
{{begin-eqn}}
{{eqn | l = \map f x
... | By definition of Fourier series:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \frac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x
}}
{{eqn | l = b_n
| r = \frac 1 \pi \int_0^{2 \pi} \map f x \sin n x \rd x
}}
... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 {2 \pi}$ as:
:[[File:Sneddon-1-Exercise-1.png|500px|thumb|right|$\map f x$ and its $7$th approximation]]
:<nowiki>$\map f x = \begin{cases}
-\pi & : 0 < x \le \pi \\
x - \pi & : \pi < x < 2 \pi \end{cases}$</nowiki>
Then its [[D... | By definition of [[Definition:Fourier Series over Range 2 Pi|Fourier series]]:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
where:
{{begin-eqn}}
{{eqn | l = a_n
| r = \frac 1 \pi \int_0^{2 \pi} \map f x \cos n x \rd x
}}
{{eqn | l = b_n
| r = \frac ... | Fourier Series/Minus Pi over 0 to Pi, x minus Pi over Pi to 2 Pi | https://proofwiki.org/wiki/Fourier_Series/Minus_Pi_over_0_to_Pi,_x_minus_Pi_over_Pi_to_2_Pi | https://proofwiki.org/wiki/Fourier_Series/Minus_Pi_over_0_to_Pi,_x_minus_Pi_over_Pi_to_2_Pi | [
"Examples of Fourier Series"
] | [
"Definition:Real Function",
"File:Sneddon-1-Exercise-1.png",
"Definition:Fourier Series/Range 2 Pi"
] | [
"Definition:Fourier Series/Range 2 Pi",
"Cosine of Zero is One",
"Linear Combination of Integrals/Definite",
"Sum of Integrals on Adjacent Intervals for Continuous Functions",
"Primitive of Constant",
"Primitive of Power",
"Linear Combination of Integrals/Definite",
"Sum of Integrals on Adjacent Inter... |
proofwiki-14128 | Rouché's Theorem | Let $\gamma$ be a closed contour.
Let $D$ be the region enclosed by $\gamma$.
Let $f$ and $g$ be complex-valued functions which are holomorphic in $D$.
Let $\cmod {\map g z} < \cmod {\map f z}$ on $\gamma$.
Then $f$ and $f + g$ have the same number of zeroes in $D$ counted up to multiplicity. | Let $N_f$ and $N_{f + g}$ be the number of zeroes of $f$ and $f + g$ in $D$ respectively.
By the Argument Principle:
:$\ds N_f = \frac 1 {2 \pi i} \oint_\gamma \frac {\map {f'} z} {\map f z} \rd z$
Similarly:
:$\ds N_{f + g} = \frac 1 {2 \pi i} \oint_\gamma \frac {\map {\paren {f + g}'} z} {\map {\paren {f + g} } z}... | Let $\gamma$ be a [[Definition:Closed Contour (Complex Plane)|closed contour]].
Let $D$ be the [[Definition:Region (Complex Analysis)|region]] enclosed by $\gamma$.
Let $f$ and $g$ be [[Definition:Complex-Valued Function|complex-valued functions]] which are [[Definition:Holomorphic Function|holomorphic]] in $D$.
L... | Let $N_f$ and $N_{f + g}$ be the number of zeroes of $f$ and $f + g$ in $D$ respectively.
By the [[Argument Principle]]:
:$\ds N_f = \frac 1 {2 \pi i} \oint_\gamma \frac {\map {f'} z} {\map f z} \rd z$
Similarly:
:$\ds N_{f + g} = \frac 1 {2 \pi i} \oint_\gamma \frac {\map {\paren {f + g}'} z} {\map {\paren {f +... | Rouché's Theorem | https://proofwiki.org/wiki/Rouché's_Theorem | https://proofwiki.org/wiki/Rouché's_Theorem | [
"Complex Analysis"
] | [
"Definition:Contour/Closed/Complex Plane",
"Definition:Region/Complex",
"Definition:Complex-Valued Function",
"Definition:Holomorphic Function",
"Definition:Root of Mapping",
"Definition:Multiplicity (Complex Analysis)"
] | [
"Argument Principle",
"Linear Combination of Contour Integrals",
"Product Rule for Derivatives",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Encircle",
"Definition:Winding Number",
"Category:Complex Analysis"
] |
proofwiki-14129 | Argument Principle | Let $\gamma$ be a closed contour.
Let $D$ be the region enclosed by $\gamma$.
Let $f$ be a function meromorphic on $D$.
Let $f$ be holomorphic with no zeroes on $\gamma$.
Let $N$ denote the number of zeroes of $f$ in $D$, counted up to multiplicity.
Let $P$ denote the number of poles of $f$ in $D$, counted up to orde... | Let $n_1, n_2, n_3 \ldots n_N$ denote the zeroes of $f$, and $p_1, p_2, p_3 \ldots p_P$ denote its poles.
Then, there exists a non-zero holomorphic function $g$ such that:
:$\ds \map f x = \frac {\prod_{k \mathop = 1}^N \paren {z - n_k} } {\prod_{j \mathop = 1}^P \paren {z - p_j} } \map g z$
Taking the logarithmic der... | Let $\gamma$ be a [[Definition:Closed Contour (Complex Plane)|closed contour]].
Let $D$ be the [[Definition:Region (Complex Analysis)|region]] enclosed by $\gamma$.
Let $f$ be a function [[Definition:Meromorphic Function|meromorphic]] on $D$.
Let $f$ be [[Definition:Holomorphic Function|holomorphic]] with no [[Def... | Let $n_1, n_2, n_3 \ldots n_N$ denote the [[Definition:Root of Mapping|zeroes]] of $f$, and $p_1, p_2, p_3 \ldots p_P$ denote its [[Definition:Pole (Complex Analysis)|poles]].
Then, there exists a non-zero [[Definition:Holomorphic Function|holomorphic function]] $g$ such that:
:$\ds \map f x = \frac {\prod_{k \matho... | Argument Principle | https://proofwiki.org/wiki/Argument_Principle | https://proofwiki.org/wiki/Argument_Principle | [
"Complex Analysis",
"Named Theorems"
] | [
"Definition:Contour/Closed/Complex Plane",
"Definition:Region/Complex",
"Definition:Meromorphic Function",
"Definition:Holomorphic Function",
"Definition:Root of Mapping",
"Definition:Root of Mapping",
"Definition:Multiplicity (Complex Analysis)",
"Definition:Isolated Singularity/Pole"
] | [
"Definition:Root of Mapping",
"Definition:Isolated Singularity/Pole",
"Definition:Holomorphic Function",
"Definition:Logarithmic Differentiation",
"Linear Combination of Contour Integrals",
"Definition:Holomorphic Function",
"Cauchy-Goursat Theorem",
"Category:Complex Analysis",
"Category:Named Theo... |
proofwiki-14130 | Ring with Unity has Prime Ideal | Let $A$ be a non-trivial commutative ring with unity.
Then $A$ has a prime ideal. | By Krull's Theorem, $A$ has a maximal ideal.
By Maximal Ideal of Commutative and Unitary Ring is Prime Ideal, $A$ has a prime ideal.
{{qed}} | Let $A$ be a [[Definition:Non-Trivial Ring|non-trivial]] [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Then $A$ has a [[Definition:Prime Ideal of Ring|prime ideal]]. | By [[Krull's Theorem]], $A$ has a [[Definition:Maximal Ideal of Ring|maximal ideal]].
By [[Maximal Ideal of Commutative and Unitary Ring is Prime Ideal]], $A$ has a [[Definition:Prime Ideal of Ring|prime ideal]].
{{qed}} | Ring with Unity has Prime Ideal | https://proofwiki.org/wiki/Ring_with_Unity_has_Prime_Ideal | https://proofwiki.org/wiki/Ring_with_Unity_has_Prime_Ideal | [
"Commutative Algebra"
] | [
"Definition:Non-Trivial Ring",
"Definition:Commutative and Unitary Ring",
"Definition:Prime Ideal of Ring"
] | [
"Krull's Theorem",
"Definition:Maximal Ideal of Ring",
"Maximal Ideal of Commutative and Unitary Ring is Prime Ideal",
"Definition:Prime Ideal of Ring"
] |
proofwiki-14131 | Proper Ideal of Ring is Contained in Maximal Ideal | Let $A$ be a commutative ring with unity.
Let $\mathfrak a \subseteq A$ be a proper ideal.
Then there exists a maximal ideal $\mathfrak m$ with $\mathfrak a \subseteq \mathfrak m$. | Let $A / \mathfrak a$ be the quotient ring.
By Proper Ideal iff Quotient Ring is Non-Null, $A / \mathfrak a$ is non-null.
By Krull's Theorem, $A / \mathfrak a$ has a maximal ideal.
By Correspondence Theorem for Quotient Rings, $A$ has a maximal ideal containing $\mathfrak a$.
{{qed}}
Category:Commutative Algebra
Catego... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\mathfrak a \subseteq A$ be a [[Definition:Proper Ideal of Ring|proper ideal]].
Then there exists a [[Definition:Maximal Ideal of Ring|maximal ideal]] $\mathfrak m$ with $\mathfrak a \subseteq \mathfrak m$. | Let $A / \mathfrak a$ be the [[Definition:Quotient Ring|quotient ring]].
By [[Proper Ideal iff Quotient Ring is Non-Null]], $A / \mathfrak a$ is [[Definition:Non-Null Ring|non-null]].
By [[Krull's Theorem]], $A / \mathfrak a$ has a [[Definition:Maximal Ideal of Ring|maximal ideal]].
By [[Correspondence Theorem for Q... | Proper Ideal of Ring is Contained in Maximal Ideal | https://proofwiki.org/wiki/Proper_Ideal_of_Ring_is_Contained_in_Maximal_Ideal | https://proofwiki.org/wiki/Proper_Ideal_of_Ring_is_Contained_in_Maximal_Ideal | [
"Commutative Algebra",
"Maximal Ideals of Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring/Proper Ideal",
"Definition:Maximal Ideal of Ring"
] | [
"Definition:Quotient Ring",
"Proper Ideal iff Quotient Ring is Non-Null",
"Definition:Non-Null Ring",
"Krull's Theorem",
"Definition:Maximal Ideal of Ring",
"Correspondence Theorem for Quotient Rings",
"Definition:Maximal Ideal of Ring",
"Category:Commutative Algebra",
"Category:Maximal Ideals of Ri... |
proofwiki-14132 | Proper Ideal iff Quotient Ring is Non-Null | Let $A$ be a commutative ring.
Let $\mathfrak a \subseteq A$ be an ideal.
{{TFAE}}
:$(1): \quad \mathfrak a$ is a proper ideal of $A$.
:$(2): \quad$ The quotient ring $A / \mathfrak a$ is a non-null ring. | === 1 implies 2 ===
Suppose $\mathfrak a$ is a proper ideal of $A$.
That is:
:$\exists x \in A \setminus \mathfrak a$
By definition of congruence modulo subgroup:
:$x + \mathfrak a \ne 0 + \mathfrak a$
in the quotient ring $A / \mathfrak a$.
Hence $A / \mathfrak a$ is a non-null ring.
{{qed|lemma}} | Let $A$ be a [[Definition:Commutative Ring|commutative ring]].
Let $\mathfrak a \subseteq A$ be an [[Definition:Ideal of Ring|ideal]].
{{TFAE}}
:$(1): \quad \mathfrak a$ is a [[Definition:Proper Ideal of Ring|proper ideal]] of $A$.
:$(2): \quad$ The [[Definition:Quotient Ring|quotient ring]] $A / \mathfrak a$ is a [[... | === 1 implies 2 ===
Suppose $\mathfrak a$ is a [[Definition:Proper Ideal of Ring|proper ideal]] of $A$.
That is:
:$\exists x \in A \setminus \mathfrak a$
By definition of [[Definition:Congruence Modulo Subgroup|congruence modulo subgroup]]:
:$x + \mathfrak a \ne 0 + \mathfrak a$
in the [[Definition:Quotient Ring|quo... | Proper Ideal iff Quotient Ring is Non-Null | https://proofwiki.org/wiki/Proper_Ideal_iff_Quotient_Ring_is_Non-Null | https://proofwiki.org/wiki/Proper_Ideal_iff_Quotient_Ring_is_Non-Null | [
"Quotient Rings"
] | [
"Definition:Commutative Ring",
"Definition:Ideal of Ring",
"Definition:Ideal of Ring/Proper Ideal",
"Definition:Quotient Ring",
"Definition:Non-Null Ring"
] | [
"Definition:Ideal of Ring/Proper Ideal",
"Definition:Congruence Modulo Subgroup",
"Definition:Quotient Ring",
"Definition:Non-Null Ring",
"Definition:Non-Null Ring",
"Definition:Congruence Modulo Subgroup",
"Definition:Ideal of Ring/Proper Ideal"
] |
proofwiki-14133 | Half-Range Fourier Sine Series/Sine of Half x over 0 to Pi, Minus Sine of Half x over Pi to 2 Pi | Let $\map f x$ be the real function defined on $\openint 0 {2 \pi}$ as:
:600pxthumbright$\map f x$ and its $7$th approximation
:<nowiki>$\map f x = \begin {cases}
\sin \dfrac x 2 & : 0 \le x < \pi \\
-\sin \dfrac x 2 & : \pi < x \le 2 \pi \end {cases}$</nowiki>
Then its Fourier series can be expressed as:
{{begin-eqn}}... | By definition of half-range Fourier sine series:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin \dfrac {n x} 2$
where:
{{begin-eqn}}
{{eqn | l = b_n
| r = \frac 2 {2 \pi} \int_0^{2 \pi} \map f x \sin \frac {n \pi x} {2 \pi} \rd x
| c =
}}
{{eqn | r = \frac 1 \pi \int_0^{2 \pi} \map f x \sin \frac... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 {2 \pi}$ as:
:[[File:Sneddon-1-Exercise-2.png|600px|thumb|right|$\map f x$ and its $7$th approximation]]
:<nowiki>$\map f x = \begin {cases}
\sin \dfrac x 2 & : 0 \le x < \pi \\
-\sin \dfrac x 2 & : \pi < x \le 2 \pi \end {cases}$... | By definition of [[Definition:Half-Range Fourier Sine Series|half-range Fourier sine series]]:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin \dfrac {n x} 2$
where:
{{begin-eqn}}
{{eqn | l = b_n
| r = \frac 2 {2 \pi} \int_0^{2 \pi} \map f x \sin \frac {n \pi x} {2 \pi} \rd x
| c =
}}
{{eqn | r ... | Half-Range Fourier Sine Series/Sine of Half x over 0 to Pi, Minus Sine of Half x over Pi to 2 Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Sine_Series/Sine_of_Half_x_over_0_to_Pi,_Minus_Sine_of_Half_x_over_Pi_to_2_Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Sine_Series/Sine_of_Half_x_over_0_to_Pi,_Minus_Sine_of_Half_x_over_Pi_to_2_Pi | [
"Examples of Half-Range Fourier Series"
] | [
"Definition:Real Function",
"File:Sneddon-1-Exercise-2.png",
"Definition:Fourier Series"
] | [
"Definition:Half-Range Fourier Sine Series",
"Primitive of Sine of a x by Sine of b x",
"Sine of Integer Multiple of Pi",
"Primitive of Square of Sine of a x",
"Sine of Integer Multiple of Pi",
"Definition:Odd Integer",
"Sine of Integer Multiple of Pi",
"Definition:Even Integer",
"Sine of Half-Integ... |
proofwiki-14134 | Residue of Quotient | Let $f$ and $g$ be functions holomorphic on some region containing $a$.
Let $g$ have a zero of multiplicity $1$ at $a$.
Then:
:$\Res {\dfrac f g} a = \dfrac {\map f a} {\map {g'} a}$
{{explain|definition of "Res", by a link to Residue}} | As $g$ has a zero of multiplicity $1$ at $a$, $\dfrac f g$ has a simple pole at $a$ by definition.
So:
{{begin-eqn}}
{{eqn | l = \Res {\dfrac f g} a
| r = \lim_{z \mathop \to a} \paren {z - a} \frac {\map f z} {\map g z}
| c = Residue at Simple Pole
}}
{{eqn | r = \lim_{z \mathop \to a} \frac {\map f z} {\frac {\map... | Let $f$ and $g$ be [[Definition:Complex Function|function]]s [[Definition:Holomorphic Function|holomorphic]] on some [[Definition:Region (Complex Analysis)|region]] containing $a$.
Let $g$ have a [[Definition:Root of Mapping|zero]] of [[Definition:Multiplicity (Complex Analysis)|multiplicity]] $1$ at $a$.
Then:
:... | As $g$ has a [[Definition:Root of Mapping|zero]] of [[Definition:Multiplicity (Complex Analysis)|multiplicity]] $1$ at $a$, $\dfrac f g$ has a [[Definition:Simple Pole|simple pole]] at $a$ by definition.
So:
{{begin-eqn}}
{{eqn | l = \Res {\dfrac f g} a
| r = \lim_{z \mathop \to a} \paren {z - a} \frac {\map f z} {... | Residue of Quotient | https://proofwiki.org/wiki/Residue_of_Quotient | https://proofwiki.org/wiki/Residue_of_Quotient | [
"Complex Analysis"
] | [
"Definition:Complex Function",
"Definition:Holomorphic Function",
"Definition:Region/Complex",
"Definition:Root of Mapping",
"Definition:Multiplicity (Complex Analysis)"
] | [
"Definition:Root of Mapping",
"Definition:Multiplicity (Complex Analysis)",
"Definition:Order of Pole/Simple Pole",
"Residue at Simple Pole",
"Combination Theorem for Limits of Functions/Complex/Quotient Rule",
"Category:Complex Analysis"
] |
proofwiki-14135 | Fourier Series/Cosine of x over Minus Pi to Zero, Minus Cosine of x over Zero to Pi | Let $\map f x$ be the real function defined on $\openint {-\pi} \pi$ as:
:600pxthumbright$\map f x$ and its $7$th approximation
:$\map f x = \begin {cases}
\cos x & : -\pi < x < 0 \\
-\cos x & : 0 < x < \pi \end {cases}$
Then its Fourier series can be expressed as:
{{begin-eqn}}
{{eqn | l = \map f x
| o = \sim
... | It is apparent by inspection that $\map f x$ is an odd function over $\openint {-\pi} \pi$.
It follows from Fourier Series for Odd Function over Symmetric Range:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin n x$
where for all $n \in \Z_{> 0}$:
:$b_n = \ds \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$
for ... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint {-\pi} \pi$ as:
:[[File:Sneddon-1-Exercise-3.png|600px|thumb|right|$\map f x$ and its $7$th approximation]]
:$\map f x = \begin {cases}
\cos x & : -\pi < x < 0 \\
-\cos x & : 0 < x < \pi \end {cases}$
Then its [[Definition:Fourier... | It is apparent by inspection that $\map f x$ is an [[Definition:Odd Function|odd function]] over $\openint {-\pi} \pi$.
It follows from [[Fourier Series for Odd Function over Symmetric Range]]:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin n x$
where for all $n \in \Z_{> 0}$:
:$b_n = \ds \frac 2 \pi \int... | Fourier Series/Cosine of x over Minus Pi to Zero, Minus Cosine of x over Zero to Pi | https://proofwiki.org/wiki/Fourier_Series/Cosine_of_x_over_Minus_Pi_to_Zero,_Minus_Cosine_of_x_over_Zero_to_Pi | https://proofwiki.org/wiki/Fourier_Series/Cosine_of_x_over_Minus_Pi_to_Zero,_Minus_Cosine_of_x_over_Zero_to_Pi | [
"Examples of Fourier Series"
] | [
"Definition:Real Function",
"File:Sneddon-1-Exercise-3.png",
"Definition:Fourier Series/Range 2 Pi"
] | [
"Definition:Odd Function",
"Fourier Series for Odd Function over Symmetric Range",
"Primitive of Sine of a x by Cosine of b x",
"Cosine of Zero is One",
"Primitive of Sine of a x by Cosine of a x",
"Sine of Integer Multiple of Pi",
"Definition:Odd Integer",
"Cosine of Integer Multiple of Pi",
"Defin... |
proofwiki-14136 | Holomorphic Function is Analytic | Let $a \in \C$ be a complex number.
Let $r > 0$ be a real number.
Let $f$ be a function holomorphic on some open disk $D = \map B {a, r}$.
Then $f$ is complex analytic on $D$. | Let $z \in D$.
Then:
{{begin-eqn}}
{{eqn | l = \map f z
| r = \frac 1 {2 \pi i} \oint_{\partial D} \frac {\map f t} {t - z} \rd t
| c = Cauchy's Integral Formula
}}
{{eqn | r = \frac 1 {2 \pi i} \oint_{\partial D} \frac {\map f t} {\paren {t - a} \paren {1 - \frac {z - a} {t - a} } } \rd t
}}
{{end-eqn}}
N... | Let $a \in \C$ be a [[Definition:Complex Number|complex number]].
Let $r > 0$ be a [[Definition:Real Number|real number]].
Let $f$ be a [[Definition:Complex Function|function]] [[Definition:Holomorphic Function|holomorphic]] on some [[Definition:Open Complex Disk|open disk]] $D = \map B {a, r}$.
Then $f$ is [[Defin... | Let $z \in D$.
Then:
{{begin-eqn}}
{{eqn | l = \map f z
| r = \frac 1 {2 \pi i} \oint_{\partial D} \frac {\map f t} {t - z} \rd t
| c = [[Cauchy's Integral Formula]]
}}
{{eqn | r = \frac 1 {2 \pi i} \oint_{\partial D} \frac {\map f t} {\paren {t - a} \paren {1 - \frac {z - a} {t - a} } } \rd t
}}
{{end-e... | Holomorphic Function is Analytic | https://proofwiki.org/wiki/Holomorphic_Function_is_Analytic | https://proofwiki.org/wiki/Holomorphic_Function_is_Analytic | [
"Holomorphic Functions",
"Analytic Complex Functions"
] | [
"Definition:Complex Number",
"Definition:Real Number",
"Definition:Complex Function",
"Definition:Holomorphic Function",
"Definition:Complex Disk/Open",
"Definition:Analytic Function/Complex Plane"
] | [
"Cauchy's Integral Formula",
"Sum of Infinite Geometric Sequence",
"Continuous Function on Compact Space is Bounded",
"Definition:Real Number",
"Definition:Real Number",
"Sum of Infinite Geometric Sequence",
"Weierstrass M-Test",
"Definition:Uniform Convergence",
"Definition:Analytic Function/Comple... |
proofwiki-14137 | Schwarz's Lemma | Let $D$ be the unit disk centred at $0$.
Let $f: D \to \C$ be a holomorphic function.
Let $\map f 0 = 0$ and $\cmod {\map f z} \le 1$ for all $z \in D$.
Then $\cmod {\map {f'} 0} \le 1$, and $\cmod {\map f z} \le \cmod z$ for all $z \in D$. | === Lemma ===
{{:Schwarz's Lemma/Lemma}}{{qed|lemma}}
Let $g: D \to \C$ be a complex function such that:
:$\map g z = \begin {cases} \dfrac {\map f z} z & z \ne 0 \\ \map {f'} 0 & z = 0 \end {cases}$
By the {{Lemma|Schwarz's Lemma}}, $g$ is holomorphic on $D$.
Let $r$ be a real number with $0 < r < 1$.
Let $C_r$ be t... | Let $D$ be the [[Definition:Unit Disk|unit disk]] centred at $0$.
Let $f: D \to \C$ be a [[Definition:Holomorphic Function|holomorphic function]].
Let $\map f 0 = 0$ and $\cmod {\map f z} \le 1$ for all $z \in D$.
Then $\cmod {\map {f'} 0} \le 1$, and $\cmod {\map f z} \le \cmod z$ for all $z \in D$. | === [[Schwarz's Lemma/Lemma|Lemma]] ===
{{:Schwarz's Lemma/Lemma}}{{qed|lemma}}
Let $g: D \to \C$ be a [[Definition:Complex Function|complex function]] such that:
:$\map g z = \begin {cases} \dfrac {\map f z} z & z \ne 0 \\ \map {f'} 0 & z = 0 \end {cases}$
By the {{Lemma|Schwarz's Lemma}}, $g$ is [[Definition:Holo... | Schwarz's Lemma | https://proofwiki.org/wiki/Schwarz's_Lemma | https://proofwiki.org/wiki/Schwarz's_Lemma | [
"Schwarz's Lemma",
"Complex Analysis"
] | [
"Definition:Unit Disk",
"Definition:Holomorphic Function"
] | [
"Schwarz's Lemma/Lemma",
"Definition:Complex Function",
"Definition:Holomorphic Function",
"Definition:Real Number",
"Definition:Complex Disk/Closed",
"Definition:Compact Space/Metric Space/Complex",
"Continuous Function on Compact Space is Bounded",
"Definition:Maximum Value of Real Function",
"Def... |
proofwiki-14138 | Ring of Integers of Number Field is Dedekind Domain | Let $K$ be an algebraic number field.
Let $\OO_K$ be its ring of integers.
Then $\OO_K$ is a Dedekind domain. | {{proof wanted}}
Category:Rings of Integers of Number Fields
Category:Dedekind Domains
Category:Algebraic Number Theory
o6i6oxc58m68oqpjj6hu6jlxxvuofok | Let $K$ be an [[Definition:Algebraic Number Field|algebraic number field]].
Let $\OO_K$ be its [[Definition:Ring of Integers of Number Field|ring of integers]].
Then $\OO_K$ is a [[Definition:Dedekind Domain|Dedekind domain]]. | {{proof wanted}}
[[Category:Rings of Integers of Number Fields]]
[[Category:Dedekind Domains]]
[[Category:Algebraic Number Theory]]
o6i6oxc58m68oqpjj6hu6jlxxvuofok | Ring of Integers of Number Field is Dedekind Domain | https://proofwiki.org/wiki/Ring_of_Integers_of_Number_Field_is_Dedekind_Domain | https://proofwiki.org/wiki/Ring_of_Integers_of_Number_Field_is_Dedekind_Domain | [
"Rings of Integers of Number Fields",
"Dedekind Domains",
"Algebraic Number Theory"
] | [
"Definition:Algebraic Number Field",
"Definition:Ring of Integers of Number Field",
"Definition:Dedekind Domain"
] | [
"Category:Rings of Integers of Number Fields",
"Category:Dedekind Domains",
"Category:Algebraic Number Theory"
] |
proofwiki-14139 | Fourier Series/Exponential of x over Minus Pi to Pi | Let $\map f x$ be the real function defined on $\R$ as:
:200pxthumbright$\map f x$ and its $7$th approximation
:$\map f x = \begin{cases}
e^x & : -\pi < x \le \pi \\
\map f {x + 2 \pi} & : \text{everywhere} \end{cases}$
Then its Fourier series can be expressed as:
{{begin-eqn}}
{{eqn | l = \map f x
| o = \sim
... | By definition of Fourier series:
:$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
where for all $n \in \Z_{> 0}$:
{{begin-eqn}}
{{eqn | l = a_n
| r = \dfrac 1 \pi \int_{-\pi}^\pi \map f x \cos n x \rd x
}}
{{eqn | l = b_n
| r = \dfrac 1 \pi \int... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\R$ as:
:[[File:Sneddon-1-Exercise-4.png|200px|thumb|right|$\map f x$ and its $7$th approximation]]
:$\map f x = \begin{cases}
e^x & : -\pi < x \le \pi \\
\map f {x + 2 \pi} & : \text{everywhere} \end{cases}$
Then its [[Definition:Fourier ... | By definition of [[Definition:Fourier Series over Range 2 Pi|Fourier series]]:
:$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
where for all $n \in \Z_{> 0}$:
{{begin-eqn}}
{{eqn | l = a_n
| r = \dfrac 1 \pi \int_{-\pi}^\pi \map f x \cos n x \rd x... | Fourier Series/Exponential of x over Minus Pi to Pi | https://proofwiki.org/wiki/Fourier_Series/Exponential_of_x_over_Minus_Pi_to_Pi | https://proofwiki.org/wiki/Fourier_Series/Exponential_of_x_over_Minus_Pi_to_Pi | [
"Examples of Fourier Series"
] | [
"Definition:Real Function",
"File:Sneddon-1-Exercise-4.png",
"Definition:Fourier Series/Range 2 Pi"
] | [
"Definition:Fourier Series/Range 2 Pi",
"Cosine of Zero is One",
"Primitive of Exponential Function",
"Primitive of Exponential of a x by Cosine of b x",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple of Pi",
"Primitive of Exponential of a x by Sine of b x",
"Sine of Integer Multiple of... |
proofwiki-14140 | Sum of Reciprocals of Squares plus 1 | :$\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^2 + 1} = \frac 1 2 \paren {\pi \coth \pi - 1}$ | Let $\map f x$ be the real function defined on $\hointl {-\pi} \pi$ as:
:$\map f x = e^x$
From Fourier Series: $e^x$ over $\openint {-\pi} \pi$, we have:
:$(1): \quad \ds \map f x \sim \map S x = \frac {\sinh \pi} \pi \paren {1 + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \paren {\frac {\cos n x} {1 + n^2} - \frac {n ... | :$\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^2 + 1} = \frac 1 2 \paren {\pi \coth \pi - 1}$ | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\hointl {-\pi} \pi$ as:
:$\map f x = e^x$
From [[Fourier Series/Exponential of x over Minus Pi to Pi|Fourier Series: $e^x$ over $\openint {-\pi} \pi$]], we have:
:$(1): \quad \ds \map f x \sim \map S x = \frac {\sinh \pi} \pi \paren {1 + 2 ... | Sum of Reciprocals of Squares plus 1 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Squares_plus_1 | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Squares_plus_1 | [
"Hyperbolic Cotangent Function"
] | [] | [
"Definition:Real Function",
"Fourier Series/Exponential of x over Minus Pi to Pi",
"Fourier's Theorem",
"Cosine of Integer Multiple of Pi",
"Sine of Integer Multiple of Pi"
] |
proofwiki-14141 | Half-Range Fourier Cosine Series/Cosine of Non-Integer Multiple of x over 0 to Pi | Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer.
:500pxthumbright$\map f x$ for $\lambda = 4 \cdotp 6$ and its $5$th approximation
Let $\map f x$ be the real function defined on $\openint 0 \pi$ as:
:$\map f x = \cos \lambda x$
Then its half-range Fourier cosine series can be expressed as:
{{... | By definition of half-range Fourier cosine series:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{> 0}$:
:$a_n = \ds \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$
Thus by definition of $f$:
{{begin-eqn}}
{{eqn | l = a_0
| r = \frac 2 \pi \int_0^\pi \map f ... | Let $\lambda \in \R \setminus \Z$ be a [[Definition:Real Number|real number]] which is not an [[Definition:Integer|integer]].
:[[File:Sneddon-1-Exercise-5.png|500px|thumb|right|$\map f x$ for $\lambda = 4 \cdotp 6$ and its $5$th approximation]]
Let $\map f x$ be the [[Definition:Real Function|real function]] defined ... | By definition of [[Definition:Half-Range Fourier Cosine Series|half-range Fourier cosine series]]:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{> 0}$:
:$a_n = \ds \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$
Thus by definition of $f$:
{{begin-eqn}}
{{eq... | Half-Range Fourier Cosine Series/Cosine of Non-Integer Multiple of x over 0 to Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Cosine_Series/Cosine_of_Non-Integer_Multiple_of_x_over_0_to_Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Cosine_Series/Cosine_of_Non-Integer_Multiple_of_x_over_0_to_Pi | [
"Examples of Half-Range Fourier Series"
] | [
"Definition:Real Number",
"Definition:Integer",
"File:Sneddon-1-Exercise-5.png",
"Definition:Real Function",
"Definition:Half-Range Fourier Cosine Series"
] | [
"Definition:Half-Range Fourier Cosine Series",
"Cosine of Zero is One",
"Primitive of Cosine Function/Corollary",
"Sine of Zero is Zero",
"Primitive of Cosine of a x by Cosine of b x",
"Sine of Integer Multiple of Pi",
"Sine of Sum",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple of P... |
proofwiki-14142 | Series Expansion for Pi Cosecant of Pi Lambda | Let $\lambda \in \R \setminus \Z$ be a real number which is not an integer.
Then:
:$\ds \pi \csc \pi \lambda = \sum_{n \mathop = 1}^\infty \paren {-1}^n \paren {\frac 1 {n + \lambda} + \frac 1 {n - 1 - \lambda} }$ | Let $\map f x$ be the real function defined on $\openint 0 \pi$ as:
:$\map f x = \cos \lambda x$
From Half-Range Fourier Cosine Series: $\cos \lambda x$ over $\openint 0 \pi$ its Fourier series can be expressed as:
:$\ds \cos \lambda x \sim \frac {2 \lambda \sin \lambda \pi} \pi \paren {\frac 1 {2 \lambda^2} + \sum_{n ... | Let $\lambda \in \R \setminus \Z$ be a [[Definition:Real Number|real number]] which is not an [[Definition:Integer|integer]].
Then:
:$\ds \pi \csc \pi \lambda = \sum_{n \mathop = 1}^\infty \paren {-1}^n \paren {\frac 1 {n + \lambda} + \frac 1 {n - 1 - \lambda} }$ | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 \pi$ as:
:$\map f x = \cos \lambda x$
From [[Half-Range Fourier Cosine Series/Cosine of Non-Integer Multiple of x over 0 to Pi|Half-Range Fourier Cosine Series: $\cos \lambda x$ over $\openint 0 \pi$]] its [[Definition:Fourier Se... | Series Expansion for Pi Cosecant of Pi Lambda | https://proofwiki.org/wiki/Series_Expansion_for_Pi_Cosecant_of_Pi_Lambda | https://proofwiki.org/wiki/Series_Expansion_for_Pi_Cosecant_of_Pi_Lambda | [
"Cosecant Function"
] | [
"Definition:Real Number",
"Definition:Integer"
] | [
"Definition:Real Function",
"Half-Range Fourier Cosine Series/Cosine of Non-Integer Multiple of x over 0 to Pi",
"Definition:Fourier Series/Range 2 Pi",
"Cosine of Zero is One",
"Difference of Two Squares",
"Power Series Expansion for Logarithm of 1 + x",
"Translation of Index Variable of Summation"
] |
proofwiki-14143 | Leading Coefficient of Product of Polynomials over Integral Domain | Let $R$ be an integral domain.
Let $f, g \in R \sqbrk x$ be polynomials.
Let $c$ and $d$ be their leading coefficients.
Then $f g$ has leading coefficient $c d$. | Let $p = \deg f$ and $q = \deg g$ be their degrees.
By Degree of Product of Polynomials over Integral Domain, $\map \deg {f g} = p + q$.
For a natural number $n \ge 0$, let:
:$a_n$ be the coefficient of the monomial $x^n$ in $f$.
:$b_n$ be the coefficient of the monomial $x^n$ in $g$.
By Coefficients of Product of Two ... | Let $R$ be an [[Definition:Integral Domain|integral domain]].
Let $f, g \in R \sqbrk x$ be [[Definition:Polynomial over Ring|polynomials]].
Let $c$ and $d$ be their [[Definition:Leading Coefficient of Polynomial|leading coefficients]].
Then $f g$ has [[Definition:Leading Coefficient of Polynomial|leading coefficien... | Let $p = \deg f$ and $q = \deg g$ be their [[Definition:Degree of Polynomial|degrees]].
By [[Degree of Product of Polynomials over Integral Domain]], $\map \deg {f g} = p + q$.
For a [[Definition:Natural Number|natural number]] $n \ge 0$, let:
:$a_n$ be the [[Definition:Coefficient of Polynomial|coefficient]] of the ... | Leading Coefficient of Product of Polynomials over Integral Domain | https://proofwiki.org/wiki/Leading_Coefficient_of_Product_of_Polynomials_over_Integral_Domain | https://proofwiki.org/wiki/Leading_Coefficient_of_Product_of_Polynomials_over_Integral_Domain | [
"Polynomial Theory"
] | [
"Definition:Integral Domain",
"Definition:Polynomial over Ring",
"Definition:Leading Coefficient of Polynomial",
"Definition:Leading Coefficient of Polynomial"
] | [
"Definition:Degree of Polynomial",
"Degree of Product of Polynomials over Ring/Corollary 2",
"Definition:Natural Numbers",
"Definition:Coefficient of Polynomial",
"Definition:Monomial of Polynomial Ring",
"Definition:Coefficient of Polynomial",
"Definition:Monomial of Polynomial Ring",
"Coefficients o... |
proofwiki-14144 | Prime Ideal of Principal Ideal Domain is Maximal | Let $D$ be a principal ideal domain whose zero is $0_D$.
Let $J \subseteq D$ be a nonzero prime ideal.
Then $J$ is maximal. | As $D$ is a principal ideal domain, every ideal of $D$ is a principal ideal $\ideal r$ generated by some $r \in D$.
So, let $\ideal p$ be an arbitrary prime ideal of $D$ generated by $p$, where $p \ne 0_R$.
As $\ideal p$ is prime, $p$ is irreducible.
{{explain|Prove the above}}
The result follows from Principal Ideal o... | Let $D$ be a [[Definition:Principal Ideal Domain|principal ideal domain]] whose [[Definition:Ring Zero|zero]] is $0_D$.
Let $J \subseteq D$ be a [[Definition:Nonzero Ideal of Ring|nonzero]] [[Definition:Prime Ideal of Commutative and Unitary Ring|prime ideal]].
Then $J$ is [[Definition:Maximal Ideal of Ring|maximal]... | As $D$ is a [[Definition:Principal Ideal Domain|principal ideal domain]], every [[Definition:Ideal of Ring|ideal]] of $D$ is a [[Definition:Principal Ideal of Ring|principal ideal]] $\ideal r$ generated by some $r \in D$.
So, let $\ideal p$ be an arbitrary [[Definition:Prime Ideal of Commutative and Unitary Ring|prime... | Prime Ideal of Principal Ideal Domain is Maximal | https://proofwiki.org/wiki/Prime_Ideal_of_Principal_Ideal_Domain_is_Maximal | https://proofwiki.org/wiki/Prime_Ideal_of_Principal_Ideal_Domain_is_Maximal | [
"Principal Ideal Domains",
"Maximal Ideals of Rings",
"Prime Ideals of Rings"
] | [
"Definition:Principal Ideal Domain",
"Definition:Ring Zero",
"Definition:Non-Null Ideal",
"Definition:Prime Ideal of Ring/Commutative and Unitary Ring",
"Definition:Maximal Ideal of Ring"
] | [
"Definition:Principal Ideal Domain",
"Definition:Ideal of Ring",
"Definition:Principal Ideal of Ring",
"Definition:Prime Ideal of Ring/Commutative and Unitary Ring",
"Definition:Principal Ideal of Ring",
"Definition:Prime Ideal of Ring/Commutative and Unitary Ring",
"Definition:Irreducible Element of Ri... |
proofwiki-14145 | Gauss's Digamma Theorem | :$\ds \map \psi {\dfrac p q} = -\gamma - \ln 2 q - \dfrac \pi 2 \map \cot {\dfrac p q \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {q / 2} - 1} \map \cos {\dfrac {2 \pi p n} q} \map \ln {\map \sin {\dfrac {\pi n} q} }$ | We now note:
{{begin-eqn}}
{{eqn | l = \map \psi {z + 1}
| r = -\gamma + \harm 1 z
| c = Digamma Function in terms of General Harmonic Number
}}
{{eqn | ll= \leadsto
| l = \map \psi {\dfrac p q}
| r = -\gamma + \harm 1 {\dfrac p q - 1}
| c = $\paren {z + 1} := \dfrac p q$
}}
{{eqn | r = -\... | :$\ds \map \psi {\dfrac p q} = -\gamma - \ln 2 q - \dfrac \pi 2 \map \cot {\dfrac p q \pi} + 2 \sum_{n \mathop = 1}^{\ceiling {q / 2} - 1} \map \cos {\dfrac {2 \pi p n} q} \map \ln {\map \sin {\dfrac {\pi n} q} }$ | We now note:
{{begin-eqn}}
{{eqn | l = \map \psi {z + 1}
| r = -\gamma + \harm 1 z
| c = [[Digamma Function in terms of General Harmonic Number]]
}}
{{eqn | ll= \leadsto
| l = \map \psi {\dfrac p q}
| r = -\gamma + \harm 1 {\dfrac p q - 1}
| c = $\paren {z + 1} := \dfrac p q$
}}
{{eqn | r ... | Gauss's Digamma Theorem | https://proofwiki.org/wiki/Gauss's_Digamma_Theorem | https://proofwiki.org/wiki/Gauss's_Digamma_Theorem | [
"Digamma Function"
] | [] | [
"Digamma Function in terms of General Harmonic Number",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Abel's Limit Theorem",
"Power Series Expansion for Logarithm of 1 - x",
"Definition:Term of Expression",
"Power Series Expansion for Logarithm of 1 - x",
"Simpson's Dissection",... |
proofwiki-14146 | Gauss's Integral Form of Digamma Function | :$\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t$ | From Extension of Harmonic Number to Non-Integer Argument, we have:
:$\map H x = \gamma + \dfrac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }$
which is equivalent to:
:$\ds \map \psi z = -\gamma + H_{z - 1}$
We aim to demonstrate:
:$\ds \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd... | :$\ds \map \psi z = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} } } \rd t$ | From [[Extension of Harmonic Number to Non-Integer Argument]], we have:
:$\map H x = \gamma + \dfrac {\map {\Gamma'} {x + 1} } {\map \Gamma {x + 1} }$
which is equivalent to:
:$\ds \map \psi z = -\gamma + H_{z - 1}$
We aim to demonstrate:
:$\ds \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-z t} } {1 - e^{-t} ... | Gauss's Integral Form of Digamma Function | https://proofwiki.org/wiki/Gauss's_Integral_Form_of_Digamma_Function | https://proofwiki.org/wiki/Gauss's_Integral_Form_of_Digamma_Function | [
"Digamma Function",
"Complex Analysis",
"Definite Integrals"
] | [] | [
"Extension of Harmonic Number to Non-Integer Argument",
"Linear Combination of Integrals",
"Linear Combination of Integrals",
"Linear Combination of Integrals",
"Definite Integral to Infinity of Reciprocal of Exponential of x minus One minus Exponential of -x over x",
"Integration by Substitution",
"Der... |
proofwiki-14147 | Rodrigues' Formula for Legendre Polynomials | :$\map {P_n} x = \dfrac 1 {2^n n!} \dfrac {\d^n} {\d x^n} \paren {x^2 - 1}^n$
where:
:$n \in \N$ is a natural number
:$P_n$ is the $n$th Legendre polynomial. | {{ProofWanted}}
{{Namedfor|Olinde Rodrigues|cat = Rodrigues}} | :$\map {P_n} x = \dfrac 1 {2^n n!} \dfrac {\d^n} {\d x^n} \paren {x^2 - 1}^n$
where:
:$n \in \N$ is a [[Definition:Natural Number|natural number]]
:$P_n$ is the $n$th [[Definition:Legendre Polynomial|Legendre polynomial]]. | {{ProofWanted}}
{{Namedfor|Olinde Rodrigues|cat = Rodrigues}} | Rodrigues' Formula for Legendre Polynomials | https://proofwiki.org/wiki/Rodrigues'_Formula_for_Legendre_Polynomials | https://proofwiki.org/wiki/Rodrigues'_Formula_for_Legendre_Polynomials | [
"Legendre Polynomials",
"Rodrigues' Formula"
] | [
"Definition:Natural Numbers",
"Definition:Legendre Polynomial"
] | [] |
proofwiki-14148 | Product of Coprime Ideals equals Intersection | Let $A$ be a commutative ring with unity.
Let $\mathfrak a, \mathfrak b \subseteq A$ be coprime ideals.
Then their product equals their intersection:
:$\mathfrak a \mathfrak b = \mathfrak a \cap \mathfrak b$ | By Intersection of Ideals of Ring contains Product:
:$\mathfrak a \mathfrak b \subseteq \mathfrak a \cap \mathfrak b$
It remains to show that $\mathfrak a \mathfrak b \supseteq \mathfrak a \cap \mathfrak b$.
Let $c \in \mathfrak a \cap \mathfrak b$.
Because $\mathfrak a$ and $\mathfrak b$ are coprime, there exist $x \i... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\mathfrak a, \mathfrak b \subseteq A$ be [[Definition:Coprime Ideals|coprime ideals]].
Then their [[Definition:Product of Ideals of Ring|product]] equals their [[Definition:Set Intersection|intersection]]:
:$\mathfrak a \mathfr... | By [[Intersection of Ideals of Ring contains Product]]:
:$\mathfrak a \mathfrak b \subseteq \mathfrak a \cap \mathfrak b$
It remains to show that $\mathfrak a \mathfrak b \supseteq \mathfrak a \cap \mathfrak b$.
Let $c \in \mathfrak a \cap \mathfrak b$.
Because $\mathfrak a$ and $\mathfrak b$ are [[Definition:Coprim... | Product of Coprime Ideals equals Intersection | https://proofwiki.org/wiki/Product_of_Coprime_Ideals_equals_Intersection | https://proofwiki.org/wiki/Product_of_Coprime_Ideals_equals_Intersection | [
"Coprime Ideals"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Coprime Ideals",
"Definition:Product of Ideals of Ring",
"Definition:Set Intersection"
] | [
"Intersection of Ideals of Ring contains Product",
"Definition:Coprime Ideals"
] |
proofwiki-14149 | Ideal Contained in Finite Union of Prime Ideals | Let $A$ be a commutative ring with unity.
Let $\mathfrak p_1, \ldots, \mathfrak p_n$ be prime ideals.
Let $\mathfrak a \subseteq \ds \bigcup_{i \mathop = 1}^n \mathfrak p_i$ be an ideal contained in their union.
Then $\mathfrak a \subseteq \mathfrak p_i$ for some $i \in \{1, \ldots, n\}$. | The proof goes by induction on $n$.
For $n = 1$, the statement is trivial.
Let $n > 1$.
Suppose that the statement holds for $n - 1$ prime ideals.
It is to be shown that the statement holds for $n$ prime ideals.
{{AimForCont}} that $\mathfrak a \nsubseteq \mathfrak p_i$ for all $i \in \set {1, \ldots, n}$.
By the induc... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\mathfrak p_1, \ldots, \mathfrak p_n$ be [[Definition:Prime Ideal of Ring|prime ideals]].
Let $\mathfrak a \subseteq \ds \bigcup_{i \mathop = 1}^n \mathfrak p_i$ be an [[Definition:Ideal of Ring|ideal]] [[Definition:Set Containm... | The proof goes by [[Principle of Mathematical Induction|induction]] on $n$.
For $n = 1$, the statement is trivial.
Let $n > 1$.
Suppose that the statement holds for $n - 1$ [[Definition:Prime Ideal of Ring|prime ideals]].
It is to be shown that the statement holds for $n$ [[Definition:Prime Ideal of Ring|prime idea... | Ideal Contained in Finite Union of Prime Ideals | https://proofwiki.org/wiki/Ideal_Contained_in_Finite_Union_of_Prime_Ideals | https://proofwiki.org/wiki/Ideal_Contained_in_Finite_Union_of_Prime_Ideals | [
"Ideal Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Prime Ideal of Ring",
"Definition:Ideal of Ring",
"Definition:Subset",
"Definition:Set Union"
] | [
"Principle of Mathematical Induction",
"Definition:Prime Ideal of Ring",
"Definition:Prime Ideal of Ring",
"Definition:Contradiction",
"Definition:Summation/Indexed",
"Definition:Product over Finite Set",
"Sum of an Element in an Ideal and an Element not in the Ideal is not in the Ideal",
"Definition:... |
proofwiki-14150 | Modulus of Gamma Function of Imaginary Number | Let $t \in \R$ be a real number.
Then:
:$\cmod {\map \Gamma {i t} } = \sqrt {\dfrac {\pi \csch \pi t} t}$
where:
:$\Gamma$ is the Gamma function
:$\csch$ is the hyperbolic cosecant function. | By Euler's Reflection Formula:
:$\map \Gamma {i t} \, \map \Gamma {1 - i t} = \pi \, \map \csc {\pi i t}$
From Gamma Difference Equation:
:$-i t \, \map \Gamma {i t} \, \map \Gamma {-i t} = \pi \, \map \csc {\pi i t}$
Then:
{{begin-eqn}}
{{eqn | l = \cmod {-i t} \cmod {\map \Gamma {i t} } \cmod {\map \Gamma {-i t} }
... | Let $t \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$\cmod {\map \Gamma {i t} } = \sqrt {\dfrac {\pi \csch \pi t} t}$
where:
:$\Gamma$ is the [[Definition:Gamma Function|Gamma function]]
:$\csch$ is the [[Definition:Hyperbolic Cosecant|hyperbolic cosecant function]]. | By [[Euler's Reflection Formula]]:
:$\map \Gamma {i t} \, \map \Gamma {1 - i t} = \pi \, \map \csc {\pi i t}$
From [[Gamma Difference Equation]]:
:$-i t \, \map \Gamma {i t} \, \map \Gamma {-i t} = \pi \, \map \csc {\pi i t}$
Then:
{{begin-eqn}}
{{eqn | l = \cmod {-i t} \cmod {\map \Gamma {i t} } \cmod {\map \Gam... | Modulus of Gamma Function of Imaginary Number | https://proofwiki.org/wiki/Modulus_of_Gamma_Function_of_Imaginary_Number | https://proofwiki.org/wiki/Modulus_of_Gamma_Function_of_Imaginary_Number | [
"Modulus of Gamma Function of Imaginary Number",
"Gamma Function",
"Hyperbolic Cosecant Function"
] | [
"Definition:Real Number",
"Definition:Gamma Function",
"Definition:Hyperbolic Cosecant"
] | [
"Euler's Reflection Formula",
"Gamma Difference Equation",
"Complex Conjugate of Gamma Function",
"Hyperbolic Sine in terms of Sine",
"Hyperbolic Sine Function is Odd",
"Definition:Complex Number",
"Definition:Square Root",
"Hyperbolic Sine Function is Odd"
] |
proofwiki-14151 | Equivalence of Definitions of Unital Associative Commutative Algebra/Correspondence | Let $B$ be a algebra over $A$ that is unital, associative and commutative.
Let $\struct {C, f}$ be a ring under $A$.
{{TFAE}}
:$(1): \quad C$ is the underlying ring of $B$ and $f: A \to C$ is the canonical homomorphism to the unital algebra $B$.
:$(2): \quad B$ is the algebra defined by $f$. | Let $\cdot: A \times B \to B$ the ring action of $B$. | Let $B$ be a [[Definition:Algebra over Ring|algebra]] over $A$ that is [[Definition:Unital Algebra|unital]], [[Definition:Associative Algebra|associative]] and [[Definition:Commutative Algebra (Abstract Algebra)|commutative]].
Let $\struct {C, f}$ be a [[Definition:Ring under Ring|ring under]] $A$.
{{TFAE}}
:$(1): \... | Let $\cdot: A \times B \to B$ the [[Definition:Linear Ring Action|ring action]] of $B$. | Equivalence of Definitions of Unital Associative Commutative Algebra/Correspondence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Unital_Associative_Commutative_Algebra/Correspondence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Unital_Associative_Commutative_Algebra/Correspondence | [
"Equivalence of Definitions of Unital Associative Commutative Algebra"
] | [
"Definition:Algebra over Ring",
"Definition:Unital Algebra",
"Definition:Associative Algebra",
"Definition:Commutative Algebra (Abstract Algebra)",
"Definition:Ring under Ring",
"Definition:Underlying Ring of Associative Algebra",
"Definition:Canonical Homomorphism from Ring to Unital Algebra",
"Defin... | [
"Definition:Linear Ring Action",
"Definition:Linear Ring Action",
"Definition:Linear Ring Action"
] |
proofwiki-14152 | Existence of Homomorphism between Localizations of Ring | Let $A$ be a commutative ring with unity.
Let $S, T \subseteq A$ be multiplicatively closed subsets.
{{TFAE}}
:$(1): \quad$ There exists an $A$-algebra homomorphism $h : A_S \to A_T$ between localizations, the '''induced homomorphism'''.
:$(2): \quad S$ is a subset of the saturation of $T$.
:$(3): \quad$ The saturatio... | Let $i : A \to A_S$ and $j : A \to A_T$ be the localization homomorphisms. | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $S, T \subseteq A$ be [[Definition:Multiplicatively Closed Subset of Ring|multiplicatively closed subsets]].
{{TFAE}}
:$(1): \quad$ There exists an $A$-[[Definition:Unital Associative Commutative Algebra Homomorphism|algebra hom... | Let $i : A \to A_S$ and $j : A \to A_T$ be the [[Definition:Localization Homomorphism of Ring|localization homomorphisms]]. | Existence of Homomorphism between Localizations of Ring | https://proofwiki.org/wiki/Existence_of_Homomorphism_between_Localizations_of_Ring | https://proofwiki.org/wiki/Existence_of_Homomorphism_between_Localizations_of_Ring | [
"Localization of Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Multiplicatively Closed Subset of Ring",
"Definition:Unital Associative Commutative Algebra Homomorphism",
"Definition:Localization of Ring",
"Definition:Induced Homomorphism between Localizations of Ring",
"Definition:Subset",
"Definition:Saturatio... | [
"Definition:Localization of Ring"
] |
proofwiki-14153 | Fourier Series/Pi Squared minus x Squared over Minus Pi to Pi | Let $\map f x$ be the real function defined on $\openint {-\pi} \pi$ as:
:400pxthumbright$\map f x$ and its $4$th approximation
:$\map f x = \pi^2 - x^2$
$f$ can be expressed as a half-range Fourier cosine series thus:
{{begin-eqn}}
{{eqn | l = \map f x
| o = \sim
| r = \frac {2 \pi^2} 3 + 4 \sum_{n \mathop... | We have that:
:$\pi^2 - \paren {-x}^2 = \pi^2 - x^2$
and so $\map f x$ is even on $\openint {-\pi} \pi$.
It follows from Fourier Series for Even Function over Symmetric Range:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{> 0}$:
:$a_n = \ds \frac 2 \pi \int_0^\pi... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint {-\pi} \pi$ as:
:[[File:Sneddon-1-Exercise-6.png|400px|thumb|right|$\map f x$ and its $4$th approximation]]
:$\map f x = \pi^2 - x^2$
$f$ can be expressed as a [[Definition:Half-Range Fourier Cosine Series|half-range Fourier cosin... | We have that:
:$\pi^2 - \paren {-x}^2 = \pi^2 - x^2$
and so $\map f x$ is [[Definition:Even Function|even]] on $\openint {-\pi} \pi$.
It follows from [[Fourier Series for Even Function over Symmetric Range]]:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{> 0... | Fourier Series/Pi Squared minus x Squared over Minus Pi to Pi | https://proofwiki.org/wiki/Fourier_Series/Pi_Squared_minus_x_Squared_over_Minus_Pi_to_Pi | https://proofwiki.org/wiki/Fourier_Series/Pi_Squared_minus_x_Squared_over_Minus_Pi_to_Pi | [
"Examples of Half-Range Fourier Series"
] | [
"Definition:Real Function",
"File:Sneddon-1-Exercise-6.png",
"Definition:Half-Range Fourier Cosine Series"
] | [
"Definition:Even Function",
"Fourier Series for Even Function over Symmetric Range",
"Cosine of Zero is One",
"Primitive of Power",
"Linear Combination of Integrals/Definite",
"Primitive of Cosine Function/Corollary",
"Sine of Integer Multiple of Pi",
"Primitive of x squared by Cosine of a x",
"Sine... |
proofwiki-14154 | Half-Range Fourier Cosine Series/Identity Function/0 to Pi | Let $\map f x$ be the real function defined on $\openint 0 \pi$ as:
:400pxthumbright$\map f x$ and its $4$th approximation
:$\map f x = x$
Then its half-range Fourier cosine series can be expressed as:
{{begin-eqn}}
{{eqn | l = x
| o = \sim
| r = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {... | By definition of half-range Fourier cosine series:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{> 0}$:
:$a_n = \ds \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$
Thus by definition of $f$:
{{begin-eqn}}
{{eqn | l = a_0
| r = \frac 2 \pi \int_0^\pi \map f ... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 \pi$ as:
:[[File:Sneddon-1-Exercise-6-ii.png|400px|thumb|right|$\map f x$ and its $4$th approximation]]
:$\map f x = x$
Then its [[Definition:Half-Range Fourier Cosine Series|half-range Fourier cosine series]] can be expressed a... | By definition of [[Definition:Half-Range Fourier Cosine Series|half-range Fourier cosine series]]:
:$\ds \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty a_n \cos n x$
where for all $n \in \Z_{> 0}$:
:$a_n = \ds \frac 2 \pi \int_0^\pi \map f x \cos n x \rd x$
Thus by definition of $f$:
{{begin-eqn}}
{{eq... | Half-Range Fourier Cosine Series/Identity Function/0 to Pi/Proof 1 | https://proofwiki.org/wiki/Half-Range_Fourier_Cosine_Series/Identity_Function/0_to_Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Cosine_Series/Identity_Function/0_to_Pi/Proof_1 | [
"Fourier Series for Identity Function"
] | [
"Definition:Real Function",
"File:Sneddon-1-Exercise-6-ii.png",
"Definition:Half-Range Fourier Cosine Series"
] | [
"Definition:Half-Range Fourier Cosine Series",
"Cosine of Zero is One",
"Primitive of Power",
"Primitive of x by Cosine of a x",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple of Pi",
"Definition:Even Integer",
"Definition:Odd Integer"
] |
proofwiki-14155 | Half-Range Fourier Cosine Series/Identity Function/0 to Pi | Let $\map f x$ be the real function defined on $\openint 0 \pi$ as:
:400pxthumbright$\map f x$ and its $4$th approximation
:$\map f x = x$
Then its half-range Fourier cosine series can be expressed as:
{{begin-eqn}}
{{eqn | l = x
| o = \sim
| r = \frac \pi 2 - \frac 4 \pi \sum_{n \mathop = 1}^\infty \frac {... | Let $\map f x: \openint 0 \lambda \to \R$ be the identity function on the open real interval $\openint 0 \lambda$:
:$\forall x \in \openint 0 \lambda: \map f x = x$
From Half-Range Fourier Cosine Series for Identity Function, the half-range Fourier cosine series for $\map f x$ can be expressed as:
{{begin-eqn}}
{{eqn |... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 \pi$ as:
:[[File:Sneddon-1-Exercise-6-ii.png|400px|thumb|right|$\map f x$ and its $4$th approximation]]
:$\map f x = x$
Then its [[Definition:Half-Range Fourier Cosine Series|half-range Fourier cosine series]] can be expressed a... | Let $\map f x: \openint 0 \lambda \to \R$ be the [[Definition:Identity Function|identity function]] on the [[Definition:Open Real Interval|open real interval]] $\openint 0 \lambda$:
:$\forall x \in \openint 0 \lambda: \map f x = x$
From [[Half-Range Fourier Cosine Series for Identity Function]], the [[Definition:Half... | Half-Range Fourier Cosine Series/Identity Function/0 to Pi/Proof 2 | https://proofwiki.org/wiki/Half-Range_Fourier_Cosine_Series/Identity_Function/0_to_Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Cosine_Series/Identity_Function/0_to_Pi/Proof_2 | [
"Fourier Series for Identity Function"
] | [
"Definition:Real Function",
"File:Sneddon-1-Exercise-6-ii.png",
"Definition:Half-Range Fourier Cosine Series"
] | [
"Definition:Identity Mapping",
"Definition:Real Interval/Open",
"Half-Range Fourier Series/Identity Function/Cosine",
"Definition:Half-Range Fourier Cosine Series"
] |
proofwiki-14156 | Existence of Homomorphism between Localizations of Ring at Elements | Let $A$ be a commutative ring with unity.
Let $f, g \in A$.
{{TFAE}}
:$(1): \quad$ There exists an $A$-algebra homomorphism $h : A_f \to A_g$ between localizations, the '''induced homomorphism'''.
:$(2): \quad f$ divides some power of $g$.
:$(3): \quad$ There is an inclusion of vanishing sets: $\map V f \subseteq \map ... | === 1 implies 2 ===
Let there exist an $A$-algebra homomorphism $A_f \to A_g$.
By Existence of Homomorphism between Localizations of Ring, $f$ is contained in the saturation of the set of powers $\set {g^n : n \in \N}$.
That is, $f$ divides some power of $g$.
{{qed|lemma}} | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $f, g \in A$.
{{TFAE}}
:$(1): \quad$ There exists an $A$-[[Definition:Unital Associative Commutative Algebra Homomorphism|algebra homomorphism]] $h : A_f \to A_g$ between [[Definition:Localization of Ring at Element|localization... | === 1 implies 2 ===
Let there exist an $A$-[[Definition:Unital Associative Commutative Algebra Homomorphism|algebra homomorphism]] $A_f \to A_g$.
By [[Existence of Homomorphism between Localizations of Ring]], $f$ is contained in the [[Definition:Saturation of Multiplicatively Closed Subset of Ring|saturation]] of th... | Existence of Homomorphism between Localizations of Ring at Elements | https://proofwiki.org/wiki/Existence_of_Homomorphism_between_Localizations_of_Ring_at_Elements | https://proofwiki.org/wiki/Existence_of_Homomorphism_between_Localizations_of_Ring_at_Elements | [
"Localization of Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Unital Associative Commutative Algebra Homomorphism",
"Definition:Localization of Ring at Element",
"Definition:Induced Homomorphism between Localizations of Ring",
"Definition:Divisor (Algebra)/Ring with Unity",
"Definition:Power of Element/Ring",
... | [
"Definition:Unital Associative Commutative Algebra Homomorphism",
"Existence of Homomorphism between Localizations of Ring",
"Definition:Saturation of Multiplicatively Closed Subset of Ring",
"Definition:Set",
"Definition:Power of Element/Ring",
"Definition:Divisor (Algebra)/Ring with Unity",
"Definitio... |
proofwiki-14157 | Hilbert's Basis Theorem for Finitely Generated Algebras | Let $A$ be a Noetherian ring.
Let $B$ be a finitely generated algebra over $A$.
Then $B$ is a Noetherian ring. | Let $\set {b_1, \ldots , b_n} \subseteq B$ be a generator of $B$.
Let $A \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomial forms over $A$ in $\sequence {X_1, \ldots ,X_n}$.
By Hilbert's basis theorem for polynomial rings, $A \sqbrk {X_1, \ldots, X_n}$ is Noetherian.
Let $\phi : A \sqbrk {X_1, \ldots, X_n} \to B$ be ... | Let $A$ be a [[Definition:Noetherian Ring|Noetherian ring]].
Let $B$ be a [[Definition:Finitely Generated Algebra|finitely generated]] [[Definition:Unital Associative Commutative Algebra|algebra]] over $A$.
Then $B$ is a [[Definition:Noetherian Ring|Noetherian ring]]. | Let $\set {b_1, \ldots , b_n} \subseteq B$ be a [[Definition:Generator of Algebra|generator]] of $B$.
Let $A \sqbrk {X_1, \ldots, X_n}$ be the [[Definition:Ring of Polynomial Forms|ring of polynomial forms]] over $A$ in $\sequence {X_1, \ldots ,X_n}$.
By [[Hilbert's Basis Theorem/Corollary|Hilbert's basis theorem fo... | Hilbert's Basis Theorem for Finitely Generated Algebras | https://proofwiki.org/wiki/Hilbert's_Basis_Theorem_for_Finitely_Generated_Algebras | https://proofwiki.org/wiki/Hilbert's_Basis_Theorem_for_Finitely_Generated_Algebras | [
"Noetherian Rings",
"Commutative Algebra"
] | [
"Definition:Noetherian Ring",
"Definition:Finitely Generated Algebra",
"Definition:Unital Associative Commutative Algebra",
"Definition:Noetherian Ring"
] | [
"Definition:Generator of Algebra",
"Definition:Ring of Polynomial Forms",
"Hilbert's Basis Theorem/Corollary",
"Definition:Noetherian Ring",
"Definition:Ring Homomorphism",
"First Isomorphism Theorem/Rings",
"Definition:Noetherian Ring",
"Quotient Ring of Noetherian Ring is Noetherian",
"Definition:... |
proofwiki-14158 | Zariski's Lemma | Let $L / k$ be a field extension.
Let $L$ be finitely generated as an algebra over $k$.
Then $L / k$ is a finite field extension. | {{MissingLinks}}
By Noether Normalization Lemma, we find a finite and injective morphism:
:$\alpha: k \sqbrk {x_1, \dotsc, x_n} \to L$
If we can prove that $n = 0$, the proof is complete.
{{AimForCont}} $n > 0$.
Then:
:$x_1 \in k \sqbrk {x_1, \dotsc, x_n}$
and:
:$\map \alpha {x_1} \ne 0$
We have that $\map \alpha {x_1}... | Let $L / k$ be a [[Definition:Field Extension|field extension]].
Let $L$ be [[Definition:Finitely Generated Algebra|finitely generated]] [[Definition:Algebra Defined by Ring Homomorphism|as an algebra]] over $k$.
Then $L / k$ is a [[Definition:Finite Field Extension|finite field extension]]. | {{MissingLinks}}
By [[Noether Normalization Lemma]], we find a finite and injective morphism:
:$\alpha: k \sqbrk {x_1, \dotsc, x_n} \to L$
If we can prove that $n = 0$, the proof is complete.
{{AimForCont}} $n > 0$.
Then:
:$x_1 \in k \sqbrk {x_1, \dotsc, x_n}$
and:
:$\map \alpha {x_1} \ne 0$
We have that $\map \a... | Zariski's Lemma | https://proofwiki.org/wiki/Zariski's_Lemma | https://proofwiki.org/wiki/Zariski's_Lemma | [
"Commutative Algebra",
"Field Extensions"
] | [
"Definition:Field Extension",
"Definition:Finitely Generated Algebra",
"Definition:Algebra Defined by Ring Homomorphism",
"Definition:Field Extension/Degree/Finite"
] | [
"Noether Normalization Lemma",
"Category:Commutative Algebra",
"Category:Field Extensions"
] |
proofwiki-14159 | Cayley-Hamilton Theorem/Matrix | Let $A$ be a commutative ring with unity.
Let $\mathbf N = \sqbrk {a_{i j} }$ be an $n \times n$ matrix with entries in $A$.
Let $\mathbf I_n$ denote the $n \times n$ unit matrix.
Let $\map {p_{\mathbf N} } x$ be the determinant $\map \det {x \cdot \mathbf I_n - \mathbf N}$.
Then:
:$\map {p_{\mathbf N} } {\mathbf N} = ... | Taking $\phi = \mathbf N$ in the proof of Cayley-Hamilton Theorem for Finitely Generated Modules we see that $\mathbf N$ satisfies:
:$\map {p_{\mathbf N} } x = \map \det {x \cdot \mathbf I_n - \mathbf N} = 0$
Take $\mathfrak a$ to be the ideal generated by the entries of $\mathbf N$.
{{qed}}
{{explain}}
{{Namedfor|Arth... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $\mathbf N = \sqbrk {a_{i j} }$ be an $n \times n$ [[Definition:Square Matrix|matrix]] with entries in $A$.
Let $\mathbf I_n$ denote the $n \times n$ [[Definition:Unit Matrix|unit matrix]].
Let $\map {p_{\mathbf N} } x$ be the ... | Taking $\phi = \mathbf N$ in the proof of [[Cayley-Hamilton Theorem for Finitely Generated Modules]] we see that $\mathbf N$ satisfies:
:$\map {p_{\mathbf N} } x = \map \det {x \cdot \mathbf I_n - \mathbf N} = 0$
Take $\mathfrak a$ to be the ideal generated by the entries of $\mathbf N$.
{{qed}}
{{explain}}
{{Namedf... | Cayley-Hamilton Theorem/Matrix | https://proofwiki.org/wiki/Cayley-Hamilton_Theorem/Matrix | https://proofwiki.org/wiki/Cayley-Hamilton_Theorem/Matrix | [
"Cayley-Hamilton Theorem",
"Linear Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Matrix/Square Matrix",
"Definition:Unit Matrix",
"Definition:Determinant/Matrix",
"Definition:Zero Matrix"
] | [
"Cayley-Hamilton Theorem/Finitely Generated Module"
] |
proofwiki-14160 | Cayley-Hamilton Theorem/Finitely Generated Module | Let $A$ be a commutative ring with unity.
Let $M$ be a finitely generated $A$-module.
Let $\mathfrak a$ be an ideal of $A$.
Let $\phi$ be an endomorphism of $M$ such that $\phi \sqbrk M \subseteq \mathfrak a M$.
Then $\phi$ satisfies an equation of the form:
:$\phi^n + a_{n - 1} \phi^{n-1} + \cdots + a_1 \phi + a_0 = 0... | Recall that the set of endomorphisms $\map {\End_A} M$ is a unitary $A$-module.
Let $R \subseteq \map {\End_A} M$ be the $A$-submudole generated by $\set {\phi^n : n \in \N}$.
That is:
:$R = \set {a_0 + a_1 \phi + \cdots + a_k \phi^k : k \in \N, a_1, \ldots, a_k \in A}$
Note that $R$ is a commutative ring with unity.
R... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $M$ be a [[Definition:Finitely Generated Module|finitely generated]] [[Definition:Module over Ring|$A$-module]].
Let $\mathfrak a$ be an [[Definition:Ideal of Ring|ideal]] of $A$.
Let $\phi$ be an [[Definition:Module Endomorphi... | Recall that the [[Definition:Set|set]] of [[Definition:Module Endomorphism|endomorphisms]] $\map {\End_A} M$ is a [[Definition:Unitary Module over Ring|unitary $A$-module]].
Let $R \subseteq \map {\End_A} M$ be the [[Definition:Generator of Unitary Module|$A$-submudole generated]] by $\set {\phi^n : n \in \N}$.
That ... | Cayley-Hamilton Theorem/Finitely Generated Module | https://proofwiki.org/wiki/Cayley-Hamilton_Theorem/Finitely_Generated_Module | https://proofwiki.org/wiki/Cayley-Hamilton_Theorem/Finitely_Generated_Module | [
"Cayley-Hamilton Theorem",
"Commutative Algebra"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Finitely Generated Module",
"Definition:Module over Ring",
"Definition:Ideal of Ring",
"Definition:Module Endomorphism"
] | [
"Definition:Set",
"Definition:Module Endomorphism",
"Definition:Unitary Module over Ring",
"Definition:Generator of Module/Unitary",
"Definition:Commutative and Unitary Ring",
"Definition:Ring of Square Matrices",
"Definition:Generator of Module",
"Definition:Kronecker_Delta",
"Definition:Adjugate M... |
proofwiki-14161 | Weak Nullstellensatz | Let $K$ be an algebraically closed field.
Let $n \ge 0$ be an natural number.
Let $K \sqbrk {x_1, \ldots, x_n}$ be the polynomial ring in $n$ variables over $k$.
Let $I \subseteq K \sqbrk {x_1,\ldots, x_n}$ be an ideal.
{{TFAE}}
# $I$ is the unit ideal: $I = (1)$.
# Its zero-locus is empty set: $\map V I = \O$. | {{proof wanted}}
Category:Algebraic Geometry
ia15k477bwcg0dqffpy95zox0hid4sq | Let $K$ be an [[Definition:Algebraically Closed Field|algebraically closed field]].
Let $n \ge 0$ be an [[Definition:Natural Number|natural number]].
Let $K \sqbrk {x_1, \ldots, x_n}$ be the [[Definition:Polynomial Ring in Multiple Variables|polynomial ring]] in $n$ variables over $k$.
Let $I \subseteq K \sqbrk {x_1... | {{proof wanted}}
[[Category:Algebraic Geometry]]
ia15k477bwcg0dqffpy95zox0hid4sq | Weak Nullstellensatz | https://proofwiki.org/wiki/Weak_Nullstellensatz | https://proofwiki.org/wiki/Weak_Nullstellensatz | [
"Algebraic Geometry"
] | [
"Definition:Algebraically Closed Field",
"Definition:Natural Numbers",
"Definition:Polynomial Ring",
"Definition:Ideal of Ring",
"Definition:Unit Ideal",
"Definition:Zero Locus of Set of Polynomials",
"Definition:Empty Set"
] | [
"Category:Algebraic Geometry"
] |
proofwiki-14162 | Finite-Dimensional Integral Domain over Field is Field | Let $k$ be a field.
Let $R$ be a $k$-algebra of finite dimension which is an integral domain.
Then $R$ is a field. | {{tidy}}
We will prove this using the Rank - Nullity Theorem applied to $A$ viewing it as a finite dimensional vector space over $K$. Pick a non-zero element $y \in A$ and consider the $K$ - linear transformation $$\begin{eqnarray*} T&\colon&A \longrightarrow A \\
&& x\mapsto yx.\end{eqnarray*} $$
Now the kernel of thi... | Let $k$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $R$ be a $k$-[[Definition:Unital Associative Commutative Algebra|algebra]] of [[Definition:Finite Dimensional Vector Space|finite dimension]] which is an [[Definition:Integral Domain|integral domain]].
Then $R$ is a [[Definition:Field (Abstract Algebra)... | {{tidy}}
We will prove this using the [[Rank_Plus_Nullity_Theorem|Rank - Nullity Theorem]] applied to $A$ viewing it as a finite dimensional vector space over $K$. Pick a non-zero element $y \in A$ and consider the $K$ - linear transformation $$\begin{eqnarray*} T&\colon&A \longrightarrow A \\
&& x\mapsto yx.\end{eqna... | Finite-Dimensional Integral Domain over Field is Field | https://proofwiki.org/wiki/Finite-Dimensional_Integral_Domain_over_Field_is_Field | https://proofwiki.org/wiki/Finite-Dimensional_Integral_Domain_over_Field_is_Field | [
"Linear Algebra",
"Integral Domains",
"Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Unital Associative Commutative Algebra",
"Definition:Dimension of Vector Space/Finite",
"Definition:Integral Domain",
"Definition:Field (Abstract Algebra)"
] | [
"Rank_Plus_Nullity_Theorem"
] |
proofwiki-14163 | Preimage of Maximal Ideal of Finitely Generated Algebra is Maximal | Let $k$ be a field.
Let $A$ and $B$ be $k$-algebras.
Let $f: A \to B$ be a $k$-algebra homomorphism.
Let $B$ be finitely generated over $k$.
Let $\mathfrak m$ be a maximal ideal of $B$.
Then its preimage $\map {f^{-1} } {\mathfrak m}$ is a maximal ideal of $A$. | We have an injective morphism:
:$\dfrac A {\map {f^{-1} } {\mathfrak m} } \to \dfrac B {\mathfrak m}$
{{explain|The exact meaning of the above line needs to be defined.}}
We have that $\dfrac B {\mathfrak m}$ is a field extension of $k$ which is finitely generated.
Thus, by Zariski's Lemma, $\dfrac B {\mathfrak m}$ is ... | Let $k$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $A$ and $B$ be $k$-[[Definition:Unital Associative Commutative Algebra|algebras]].
Let $f: A \to B$ be a $k$-[[Definition:Unital Associative Commutative Algebra Homomorphism|algebra homomorphism]].
Let $B$ be [[Definition:Finitely Generated Algebra|fini... | We have an [[Definition:Injection|injective]] [[Definition:Unital Associative Commutative Algebra Homomorphism|morphism]]:
:$\dfrac A {\map {f^{-1} } {\mathfrak m} } \to \dfrac B {\mathfrak m}$
{{explain|The exact meaning of the above line needs to be defined.}}
We have that $\dfrac B {\mathfrak m}$ is a [[Definitio... | Preimage of Maximal Ideal of Finitely Generated Algebra is Maximal | https://proofwiki.org/wiki/Preimage_of_Maximal_Ideal_of_Finitely_Generated_Algebra_is_Maximal | https://proofwiki.org/wiki/Preimage_of_Maximal_Ideal_of_Finitely_Generated_Algebra_is_Maximal | [
"Commutative Algebra",
"Algebraic Geometry"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Unital Associative Commutative Algebra",
"Definition:Unital Associative Commutative Algebra Homomorphism",
"Definition:Finitely Generated Algebra",
"Definition:Maximal Ideal of Ring",
"Definition:Preimage/Mapping/Subset",
"Definition:Maximal Ideal of Ri... | [
"Definition:Injection",
"Definition:Unital Associative Commutative Algebra Homomorphism",
"Definition:Field Extension",
"Definition:Finitely Generated Algebra",
"Zariski's Lemma",
"Definition:Field Extension/Degree/Finite",
"Unital Subalgebra of Finite Field Extension is Field",
"Definition:Field (Abs... |
proofwiki-14164 | Unital Subalgebra of Algebraic Field Extension is Field | Let $E / F$ be an algebraic field extension.
Let $A \subseteq E$ be a unital subalgebra over $F$.
Then $A$ is a field. | By Integral Ring Extension is Integral over Intermediate Ring, $E$ is integral over $A$.
Let $a \in A$ be nonzero.
Because $E$ is a field, $a$ is a unit of $E$.
By Ring Element is Unit iff Unit in Integral Extension, $a$ is a unit of $A$.
Thus $A$ is a field.
{{qed}} | Let $E / F$ be an [[Definition:Algebraic Field Extension|algebraic field extension]].
Let $A \subseteq E$ be a [[Definition:Unital Subalgebra|unital subalgebra]] over $F$.
Then $A$ is a [[Definition:Field (Abstract Algebra)|field]]. | By [[Integral Ring Extension is Integral over Intermediate Ring]], $E$ is [[Definition:Integral Ring Extension|integral]] over $A$.
Let $a \in A$ be [[Definition:Nonzero Ring Element|nonzero]].
Because $E$ is a [[Definition:Field (Abstract Algebra)|field]], $a$ is a [[Definition:Unit of Ring|unit]] of $E$.
By [[Ring... | Unital Subalgebra of Algebraic Field Extension is Field | https://proofwiki.org/wiki/Unital_Subalgebra_of_Algebraic_Field_Extension_is_Field | https://proofwiki.org/wiki/Unital_Subalgebra_of_Algebraic_Field_Extension_is_Field | [
"Unital Subalgebras",
"Algebraic Field Extensions"
] | [
"Definition:Algebraic Field Extension",
"Definition:Unital Subalgebra",
"Definition:Field (Abstract Algebra)"
] | [
"Integral Ring Extension is Integral over Intermediate Ring",
"Definition:Integral Ring Extension",
"Definition:Nonzero Ring Element",
"Definition:Field (Abstract Algebra)",
"Definition:Unit of Ring",
"Ring Element is Unit iff Unit in Integral Extension",
"Definition:Unit of Ring",
"Definition:Field (... |
proofwiki-14165 | Unital Subalgebra of Finite Field Extension is Field | Let $E / F$ be a finite field extension.
Let $A \subseteq E$ be a unital subalgebra over $F$.
Then $A$ is a field. | By Finite Field Extension is Algebraic, $E/F$ is algebraic.
The result follows from Unital Subalgebra of Algebraic Field Extension is Field
{{qed}} | Let $E / F$ be a [[Definition:Finite Field Extension|finite field extension]].
Let $A \subseteq E$ be a [[Definition:Unital Subalgebra|unital subalgebra]] over $F$.
Then $A$ is a [[Definition:Field (Abstract Algebra)|field]]. | By [[Finite Field Extension is Algebraic]], $E/F$ is [[Definition:Algebraic Field Extension|algebraic]].
The result follows from [[Unital Subalgebra of Algebraic Field Extension is Field]]
{{qed}} | Unital Subalgebra of Finite Field Extension is Field/Proof 2 | https://proofwiki.org/wiki/Unital_Subalgebra_of_Finite_Field_Extension_is_Field | https://proofwiki.org/wiki/Unital_Subalgebra_of_Finite_Field_Extension_is_Field/Proof_2 | [
"Unital Subalgebra of Finite Field Extension is Field",
"Unital Subalgebras",
"Finite Field Extensions"
] | [
"Definition:Field Extension/Degree/Finite",
"Definition:Unital Subalgebra",
"Definition:Field (Abstract Algebra)"
] | [
"Finite Field Extension is Algebraic",
"Definition:Algebraic Field Extension",
"Unital Subalgebra of Algebraic Field Extension is Field"
] |
proofwiki-14166 | Piecewise Continuous Function with Improper Integrals may not be Bounded | Let $f$ be a real function defined on a closed interval $\closedint a b$, $a < b$.
Let $f$ be a piecewise continuous function with improper integrals.
Then $f$ may not be piecewise continuous and bounded on $\closedint a b$. | Consider the function:
:$\map f x = \begin{cases}
0 & : x = a \\
\dfrac 1 {\sqrt{x - a} } & : x \in \hointl a b
\end{cases}$
Since $\dfrac 1 {\sqrt{x - a} }$ is continuous on $\openint a b$, $f$ is continuous on $\openint a b$.
Therefore, $f$ satisfies $(1)$ in the requirements of a piecewise continuous function with i... | Let $f$ be a [[Definition:Real Function|real function]] defined on a [[Definition:Closed Real Interval|closed interval]] $\closedint a b$, $a < b$.
Let $f$ be a [[Definition:Piecewise Continuous Function with Improper Integrals|piecewise continuous function with improper integrals]].
Then $f$ may not be [[Definitio... | Consider the [[Definition:Real Function|function]]:
:$\map f x = \begin{cases}
0 & : x = a \\
\dfrac 1 {\sqrt{x - a} } & : x \in \hointl a b
\end{cases}$
Since $\dfrac 1 {\sqrt{x - a} }$ is [[Definition:Continuous Real Function on Open Interval|continuous]] on $\openint a b$, $f$ is [[Definition:Continuous Real Funct... | Piecewise Continuous Function with Improper Integrals may not be Bounded | https://proofwiki.org/wiki/Piecewise_Continuous_Function_with_Improper_Integrals_may_not_be_Bounded | https://proofwiki.org/wiki/Piecewise_Continuous_Function_with_Improper_Integrals_may_not_be_Bounded | [
"Piecewise Continuous Functions"
] | [
"Definition:Real Function",
"Definition:Real Interval/Closed",
"Definition:Piecewise Continuous Function/Improper Integrals",
"Definition:Piecewise Continuous Function/Bounded"
] | [
"Definition:Real Function",
"Definition:Continuous Real Function/Open Interval",
"Definition:Continuous Real Function/Open Interval",
"Definition:Piecewise Continuous Function/Improper Integrals",
"Definition:Subdivision of Interval",
"Definition:Continuous Real Function/Left-Continuous",
"Definition:Co... |
proofwiki-14167 | Weierstrass's Elliptic Function is Even in z | Let $\omega_1$ and $\omega_2$ be non-zero complex constants with $\dfrac {\omega_1} {\omega_2}$ having a positive imaginary part.
For $z \in \C \setminus \set {2 m \omega_1 + 2 n \omega_2: \tuple {n, m} \in \Z^2}$:
:$\ds \map \wp {-z; \omega_1, \omega_2} = \map \wp {z; \omega_1, \omega_2}$
That is, Weierstrass's ellip... | {{begin-eqn}}
{{eqn | l = \map \wp {-z; \omega_1, \omega_2}
| r = \frac 1 {\paren {-z}^2} + \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \paren {\frac 1 {\paren {-z - 2 m \omega_1 - 2 n \omega_2}^2} - \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^2} }
| c = {{Defof|Weierstrass's Elliptic Function}}
... | Let $\omega_1$ and $\omega_2$ be non-zero complex constants with $\dfrac {\omega_1} {\omega_2}$ having a [[Definition:Positive Number|positive]] [[Definition:Imaginary Part|imaginary part]].
For $z \in \C \setminus \set {2 m \omega_1 + 2 n \omega_2: \tuple {n, m} \in \Z^2}$:
:$\ds \map \wp {-z; \omega_1, \omega_2} =... | {{begin-eqn}}
{{eqn | l = \map \wp {-z; \omega_1, \omega_2}
| r = \frac 1 {\paren {-z}^2} + \sum_{\tuple {n, m} \mathop \in \Z^2 \setminus \tuple {0, 0} } \paren {\frac 1 {\paren {-z - 2 m \omega_1 - 2 n \omega_2}^2} - \frac 1 {\paren {2 m \omega_1 + 2 n \omega_2}^2} }
| c = {{Defof|Weierstrass's Elliptic Function}}
... | Weierstrass's Elliptic Function is Even in z | https://proofwiki.org/wiki/Weierstrass's_Elliptic_Function_is_Even_in_z | https://proofwiki.org/wiki/Weierstrass's_Elliptic_Function_is_Even_in_z | [
"Weierstrass's Elliptic Function"
] | [
"Definition:Positive/Number",
"Definition:Complex Number/Imaginary Part",
"Definition:Weierstrass's Elliptic Function",
"Definition:Even Function"
] | [] |
proofwiki-14168 | Modulus of Gamma Function of One Half plus Imaginary Number | Let $t \in \R$ be a real number.
Then:
:$\cmod {\map \Gamma {\dfrac 1 2 + i t} } = \sqrt {\pi \map \sech {\pi t} }$
where:
:$\Gamma$ is the Gamma function
:$\sech$ is the hyperbolic secant function. | {{begin-eqn}}
{{eqn | l = \cmod {\map \Gamma {\frac 1 2 + i t} }^2
| r = \map \Gamma {\frac 1 2 + i t} \overline {\map \Gamma {\frac 1 2 + i t} }
| c = Modulus in Terms of Conjugate
}}
{{eqn | r = \map \Gamma {\frac 1 2 + i t} \map \Gamma {\frac 1 2 - i t}
| c = Complex Conjugate of Gamma Function
}}
{{eqn | r = \ma... | Let $t \in \R$ be a [[Definition:Real Number|real number]].
Then:
:$\cmod {\map \Gamma {\dfrac 1 2 + i t} } = \sqrt {\pi \map \sech {\pi t} }$
where:
:$\Gamma$ is the [[Definition:Gamma Function|Gamma function]]
:$\sech$ is the [[Definition:Hyperbolic Secant|hyperbolic secant function]]. | {{begin-eqn}}
{{eqn | l = \cmod {\map \Gamma {\frac 1 2 + i t} }^2
| r = \map \Gamma {\frac 1 2 + i t} \overline {\map \Gamma {\frac 1 2 + i t} }
| c = [[Modulus in Terms of Conjugate]]
}}
{{eqn | r = \map \Gamma {\frac 1 2 + i t} \map \Gamma {\frac 1 2 - i t}
| c = [[Complex Conjugate of Gamma Function]]
}}
{{eqn |... | Modulus of Gamma Function of One Half plus Imaginary Number | https://proofwiki.org/wiki/Modulus_of_Gamma_Function_of_One_Half_plus_Imaginary_Number | https://proofwiki.org/wiki/Modulus_of_Gamma_Function_of_One_Half_plus_Imaginary_Number | [
"Gamma Function",
"Hyperbolic Secant Function"
] | [
"Definition:Real Number",
"Definition:Gamma Function",
"Definition:Hyperbolic Secant"
] | [
"Modulus in Terms of Conjugate",
"Complex Conjugate of Gamma Function",
"Euler's Reflection Formula",
"Sine of Complement equals Cosine",
"Hyperbolic Cosine in terms of Cosine",
"Definition:Complex Number",
"Definition:Square Root",
"Category:Gamma Function",
"Category:Hyperbolic Secant Function"
] |
proofwiki-14169 | Integral to Infinity of Sine p x over x | :$\ds \int_0^\infty \frac {\sin p x} x \rd x = \begin {cases} \dfrac \pi 2 & : p > 0 \\ \\ 0 & : p = 0 \\ \\ -\dfrac \pi 2 & : p < 0 \end {cases}$ | Let $p > 0$.
We have:
{{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sin p x} x \rd x
| r = \frac 1 p \int_0^\infty \frac {\sin t} { \frac 1 p t} \rd t
| c = substituting $t = p x$
}}
{{eqn | r = \int_0^\infty \frac {\sin t} t \rd t
}}
{{eqn | r = \frac \pi 2
| c = Dirichlet Integral
}}
{{end-eqn}}
Then:
{{begin-e... | :$\ds \int_0^\infty \frac {\sin p x} x \rd x = \begin {cases} \dfrac \pi 2 & : p > 0 \\ \\ 0 & : p = 0 \\ \\ -\dfrac \pi 2 & : p < 0 \end {cases}$ | Let $p > 0$.
We have:
{{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sin p x} x \rd x
| r = \frac 1 p \int_0^\infty \frac {\sin t} { \frac 1 p t} \rd t
| c = [[Integration by Substitution|substituting]] $t = p x$
}}
{{eqn | r = \int_0^\infty \frac {\sin t} t \rd t
}}
{{eqn | r = \frac \pi 2
| c = [[Dirichlet Int... | Integral to Infinity of Sine p x over x/Proof 1 | https://proofwiki.org/wiki/Integral_to_Infinity_of_Sine_p_x_over_x | https://proofwiki.org/wiki/Integral_to_Infinity_of_Sine_p_x_over_x/Proof_1 | [
"Definite Integrals involving Sine Function",
"Integral to Infinity of Sine p x over x"
] | [] | [
"Integration by Substitution",
"Dirichlet Integral",
"Sine Function is Odd",
"Sine of Zero is Zero"
] |
proofwiki-14170 | Integral to Infinity of Sine p x over x | :$\ds \int_0^\infty \frac {\sin p x} x \rd x = \begin {cases} \dfrac \pi 2 & : p > 0 \\ \\ 0 & : p = 0 \\ \\ -\dfrac \pi 2 & : p < 0 \end {cases}$ | {{ProofWanted|Use Primitive of Sine of a x over x, and also the analytic solution as found in {{BookLink|Integration|R.P. Gillespie|ed = 2nd|edpage = Second Edition}} }} | :$\ds \int_0^\infty \frac {\sin p x} x \rd x = \begin {cases} \dfrac \pi 2 & : p > 0 \\ \\ 0 & : p = 0 \\ \\ -\dfrac \pi 2 & : p < 0 \end {cases}$ | {{ProofWanted|Use [[Primitive of Sine of a x over x]], and also the analytic solution as found in {{BookLink|Integration|R.P. Gillespie|ed = 2nd|edpage = Second Edition}} }} | Integral to Infinity of Sine p x over x/Proof 2 | https://proofwiki.org/wiki/Integral_to_Infinity_of_Sine_p_x_over_x | https://proofwiki.org/wiki/Integral_to_Infinity_of_Sine_p_x_over_x/Proof_2 | [
"Definite Integrals involving Sine Function",
"Integral to Infinity of Sine p x over x"
] | [] | [
"Primitive of Sine of a x over x"
] |
proofwiki-14171 | Radical of Sum of Ideals | Let $A$ be a commutative ring with unity.
Let $\mathfrak a, \mathfrak b \subseteq A$ be ideals.
Then for the radical of their sum we have:
:$\map \Rad {\mathfrak a + \mathfrak b} = \map \Rad {\map \Rad {\mathfrak a} + \map \Rad {\mathfrak b} }$ | From Ideal of Ring is Contained in Radical:
:$\mathfrak a \subseteq \map \Rad {\mathfrak a}$
From Sum of Larger Ideals is Larger:
:$\mathfrak a + \mathfrak b \subseteq \map \Rad {\mathfrak a} + \map \Rad {\mathfrak b}$
From Radical of Ideal Preserves Inclusion:
:$\map \Rad {\mathfrak a + \mathfrak b} \subseteq \map \Ra... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\mathfrak a, \mathfrak b \subseteq A$ be [[Definition:Ideal of Ring|ideals]].
Then for the [[Definition:Radical of Ideal of Ring|radical]] of their [[Definition:Sum of Ideals of Ring|sum]] we have:
:$\map \Rad {\mathfrak a + \m... | From [[Ideal of Ring is Contained in Radical]]:
:$\mathfrak a \subseteq \map \Rad {\mathfrak a}$
From [[Sum of Larger Ideals is Larger]]:
:$\mathfrak a + \mathfrak b \subseteq \map \Rad {\mathfrak a} + \map \Rad {\mathfrak b}$
From [[Radical of Ideal Preserves Inclusion]]:
:$\map \Rad {\mathfrak a + \mathfrak b} \sub... | Radical of Sum of Ideals | https://proofwiki.org/wiki/Radical_of_Sum_of_Ideals | https://proofwiki.org/wiki/Radical_of_Sum_of_Ideals | [
"Radical of Ideals"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Radical of Ideal of Ring",
"Definition:Sum of Ideals of Ring"
] | [
"Ideal of Ring is Contained in Radical",
"Sum of Larger Ideals is Larger",
"Radical of Ideal Preserves Inclusion",
"Definition:Power of Element/Ring",
"Definition:Sum of Ideals of Ring",
"Binomial Theorem",
"Definition:Subset",
"Definition:Set Equality",
"Category:Radical of Ideals"
] |
proofwiki-14172 | Ideal of Ring is Contained in Radical | Let $A$ be a commutative ring with unity.
Let $\mathfrak a \subseteq A$ be an ideal.
Then $\mathfrak a$ is contained in its radical:
:$\mathfrak a \subseteq \operatorname{Rad} \left({\mathfrak a}\right)$ | Let $a \in \mathfrak a$.
By definition of power:
: $a^1 = a$
Thus:
: $a \in \operatorname{Rad} \left({\mathfrak a}\right)$
{{qed}}
Category:Radical of Ideals
893sgr06yi4pmfepysm4qb70v66010g | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\mathfrak a \subseteq A$ be an [[Definition:Ideal of Ring|ideal]].
Then $\mathfrak a$ is [[Definition:Set Containment|contained]] in its [[Definition:Radical of Ideal of Ring|radical]]:
:$\mathfrak a \subseteq \operatorname{Rad... | Let $a \in \mathfrak a$.
By definition of [[Definition:Power of Ring Element|power]]:
: $a^1 = a$
Thus:
: $a \in \operatorname{Rad} \left({\mathfrak a}\right)$
{{qed}}
[[Category:Radical of Ideals]]
893sgr06yi4pmfepysm4qb70v66010g | Ideal of Ring is Contained in Radical | https://proofwiki.org/wiki/Ideal_of_Ring_is_Contained_in_Radical | https://proofwiki.org/wiki/Ideal_of_Ring_is_Contained_in_Radical | [
"Radical of Ideals"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Subset",
"Definition:Radical of Ideal of Ring"
] | [
"Definition:Power of Element/Ring",
"Category:Radical of Ideals"
] |
proofwiki-14173 | Radical of Ideal Preserves Inclusion | Let $A$ be a commutative ring with unity.
Let $\mathfrak a \subseteq \mathfrak b \subseteq A$ be an ideals.
Then we have an inclusion of their radicals:
:$\operatorname{Rad} \left({\mathfrak a}\right) \subseteq \operatorname{Rad} \left({\mathfrak b}\right)$ | Let $x \in \operatorname{Rad} \left({\mathfrak a}\right)$.
We show that $x \in \operatorname{Rad} \left({\mathfrak b}\right)$.
By definition of radical, there exists $n \in \N$ such that the power $x^n \in \mathfrak a$.
Because $\mathfrak a \subseteq \mathfrak b$, also $x^n \in \mathfrak b$.
Thus $x \in \operatorname{R... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\mathfrak a \subseteq \mathfrak b \subseteq A$ be an [[Definition:Ideal of Ring|ideals]].
Then we have an [[Definition:Set Inclusion|inclusion]] of their [[Definition:Radical of Ideal of Ring|radicals]]:
:$\operatorname{Rad} \l... | Let $x \in \operatorname{Rad} \left({\mathfrak a}\right)$.
We show that $x \in \operatorname{Rad} \left({\mathfrak b}\right)$.
By definition of [[Definition:Radical of Ideal of Ring|radical]], there exists $n \in \N$ such that the [[Definition:Power of Ring Element|power]] $x^n \in \mathfrak a$.
Because $\mathfrak a... | Radical of Ideal Preserves Inclusion | https://proofwiki.org/wiki/Radical_of_Ideal_Preserves_Inclusion | https://proofwiki.org/wiki/Radical_of_Ideal_Preserves_Inclusion | [
"Radical of Ideals"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Subset",
"Definition:Radical of Ideal of Ring"
] | [
"Definition:Radical of Ideal of Ring",
"Definition:Power of Element/Ring",
"Category:Radical of Ideals"
] |
proofwiki-14174 | Ideals with Coprime Radicals are Coprime | Let $A$ be a commutative ring with unity.
Let $\mathfrak a, \mathfrak b \subseteq A$ be ideals.
Let their radicals be coprime:
:$\map \Rad {\mathfrak a} + \map \Rad {\mathfrak b} = \ideal 1$
Then $\mathfrak a$ and $\mathfrak b$ are coprime:
:$\mathfrak a + \mathfrak b = \ideal 1$ | We have:
{{begin-eqn}}
{{eqn | l = \map \Rad {\mathfrak a + \mathfrak b}
| r = \map \Rad {\map \Rad {\mathfrak a} + \map \Rad {\mathfrak b} }
| c = Radical of Sum of Ideals
}}
{{eqn | r = \map \Rad {\ideal 1}
| c = $\map \Rad {\mathfrak a}$ and $\map \Rad {\mathfrak b}$ are coprime
}}
{{eqn | r = \ide... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\mathfrak a, \mathfrak b \subseteq A$ be [[Definition:Ideal of Ring|ideals]].
Let their [[Definition:Radical of Ideal of Ring|radicals]] be [[Definition:Coprime Ideals|coprime]]:
:$\map \Rad {\mathfrak a} + \map \Rad {\mathfrak ... | We have:
{{begin-eqn}}
{{eqn | l = \map \Rad {\mathfrak a + \mathfrak b}
| r = \map \Rad {\map \Rad {\mathfrak a} + \map \Rad {\mathfrak b} }
| c = [[Radical of Sum of Ideals]]
}}
{{eqn | r = \map \Rad {\ideal 1}
| c = $\map \Rad {\mathfrak a}$ and $\map \Rad {\mathfrak b}$ are [[Definition:Coprime Id... | Ideals with Coprime Radicals are Coprime | https://proofwiki.org/wiki/Ideals_with_Coprime_Radicals_are_Coprime | https://proofwiki.org/wiki/Ideals_with_Coprime_Radicals_are_Coprime | [
"Radical of Ideals",
"Coprime Ideals"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ideal of Ring",
"Definition:Radical of Ideal of Ring",
"Definition:Coprime Ideals",
"Definition:Coprime Ideals"
] | [
"Radical of Sum of Ideals",
"Definition:Coprime Ideals",
"Radical of Unit Ideal",
"Unit Ideal iff Radical is Unit Ideal",
"Definition:Coprime Ideals",
"Category:Radical of Ideals",
"Category:Coprime Ideals"
] |
proofwiki-14175 | Radical of Unit Ideal | Let $A$ be a commutative ring with unity.
Let $\ideal 1$ be its unit ideal.
Then its radical equals $\ideal 1$:
:$\map \Rad {\ideal 1} = \ideal 1$. | By definition of ideal:
:$\map \Rad {\ideal 1} \subseteq A$
By Ideal of Ring is Contained in Radical:
:$\ideal 1 = A \subseteq \Rad {\ideal 1}$.
By definition of set equality:
:$\map \Rad {\ideal 1} = \ideal 1$
{{qed}}
Category:Radical of Ideals
hqnjo3hlzuix5y1p4fx9jk3cm5rh1u1 | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\ideal 1$ be its [[Definition:Unit Ideal|unit ideal]].
Then its [[Definition:Radical of Ideal of Ring|radical]] equals $\ideal 1$:
:$\map \Rad {\ideal 1} = \ideal 1$. | By definition of [[Definition:Ideal of Ring|ideal]]:
:$\map \Rad {\ideal 1} \subseteq A$
By [[Ideal of Ring is Contained in Radical]]:
:$\ideal 1 = A \subseteq \Rad {\ideal 1}$.
By definition of [[Definition:Set Equality|set equality]]:
:$\map \Rad {\ideal 1} = \ideal 1$
{{qed}}
[[Category:Radical of Ideals]]
hqnjo3... | Radical of Unit Ideal | https://proofwiki.org/wiki/Radical_of_Unit_Ideal | https://proofwiki.org/wiki/Radical_of_Unit_Ideal | [
"Radical of Ideals"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Unit Ideal",
"Definition:Radical of Ideal of Ring"
] | [
"Definition:Ideal of Ring",
"Ideal of Ring is Contained in Radical",
"Definition:Set Equality",
"Category:Radical of Ideals"
] |
proofwiki-14176 | Radical of Power of Prime Ideal | Let $A$ be a commutative ring with unity.
Let $\mathfrak p \subseteq A$ be a prime ideal.
Let $n > 0$ be a natural number.
Then the radical of the $n$th power of $\mathfrak p$ equals $\mathfrak p$:
:$\map {\operatorname{Rad} } {\mathfrak p^n} = \mathfrak p$ | $(\subseteq):$
Let $a \in \map {\operatorname{Rad} } {\mathfrak p^n}$.
Then by definition, $a^k \in \mathfrak p^n$ for some integer $k$.
By Power of Ideal is Subset, $\mathfrak p^n \subseteq \mathfrak p$.
Hence, $a^k \in \mathfrak p$.
By Power in Prime Ideal, $a \in \mathfrak p$.
{{qed|lemma}}
$(\supseteq):$
Let $b \in... | Let $A$ be a [[Definition:Commutative Ring with Unity|commutative ring with unity]].
Let $\mathfrak p \subseteq A$ be a [[Definition:Prime Ideal of Ring|prime ideal]].
Let $n > 0$ be a [[Definition:Natural Number|natural number]].
Then the [[Definition:Radical of Ideal of Ring|radical]] of the $n$th [[Definition:Po... | $(\subseteq):$
Let $a \in \map {\operatorname{Rad} } {\mathfrak p^n}$.
Then by definition, $a^k \in \mathfrak p^n$ for some [[Definition:Integer|integer]] $k$.
By [[Power of Ideal is Subset]], $\mathfrak p^n \subseteq \mathfrak p$.
Hence, $a^k \in \mathfrak p$.
By [[Power in Prime Ideal]], $a \in \mathfrak p$.
{{q... | Radical of Power of Prime Ideal | https://proofwiki.org/wiki/Radical_of_Power_of_Prime_Ideal | https://proofwiki.org/wiki/Radical_of_Power_of_Prime_Ideal | [
"Radical of Ideals",
"Ideal Theory",
"Commutative Algebra",
"Ring Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Prime Ideal of Ring",
"Definition:Natural Numbers",
"Definition:Radical of Ideal of Ring",
"Definition:Power of Ideal of Ring"
] | [
"Definition:Integer",
"Power of Ideal is Subset",
"Power in Prime Ideal",
"Category:Radical of Ideals",
"Category:Ideal Theory",
"Category:Commutative Algebra",
"Category:Ring Theory"
] |
proofwiki-14177 | Nilpotent Ring Element plus Unity is Unit | Let $A$ be a ring with unity.
Let $1 \in A$ be its unity.
Let $a \in A$ be nilpotent.
Then $1 + a$ is a unit of $A$. | Because $a$ is nilpotent, there exists a natural number $n > 0$ with $a^n = 0$.
By Sum of Geometric Sequence in Ring:
:$\paren {1 + a} \cdot \ds \sum_{k \mathop = 0}^{n - 1} \paren {-a}^k = 1 + \paren {-a}^n$
:$\paren {\ds \sum_{k \mathop = 0}^{n - 1} \paren {-a}^k} \cdot \paren {1 + a} = 1 + \paren {-a}^n$
where $\sum... | Let $A$ be a [[Definition:Ring with Unity|ring with unity]].
Let $1 \in A$ be its [[Definition:Unity of Ring|unity]].
Let $a \in A$ be [[Definition:Nilpotent Ring Element|nilpotent]].
Then $1 + a$ is a [[Definition:Unit of Ring|unit]] of $A$. | Because $a$ is [[Definition:Nilpotent Ring Element|nilpotent]], there exists a [[Definition:Natural Number|natural number]] $n > 0$ with $a^n = 0$.
By [[Sum of Geometric Sequence in Ring]]:
:$\paren {1 + a} \cdot \ds \sum_{k \mathop = 0}^{n - 1} \paren {-a}^k = 1 + \paren {-a}^n$
:$\paren {\ds \sum_{k \mathop = 0}^{n ... | Nilpotent Ring Element plus Unity is Unit | https://proofwiki.org/wiki/Nilpotent_Ring_Element_plus_Unity_is_Unit | https://proofwiki.org/wiki/Nilpotent_Ring_Element_plus_Unity_is_Unit | [
"Nilpotent Ring Elements"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Nilpotent Ring Element",
"Definition:Unit of Ring"
] | [
"Definition:Nilpotent Ring Element",
"Definition:Natural Numbers",
"Sum of Geometric Sequence in Ring",
"Definition:Summation/Indexed",
"Negative of Nilpotent Ring Element",
"Definition:Unit of Ring"
] |
proofwiki-14178 | Ideal is Contained in Contraction of Extension | Let $A$ and $B$ be commutative rings with unity.
Let $f : A \to B$ be a ring homomorphism.
Let $\mathfrak a \subseteq A$ be an ideal.
Then $\mathfrak a$ is contained in the contraction of its extension by $f$:
:$\mathfrak a \subseteq \mathfrak a^{ec}$ | By definition of extension and generated ideal:
:$f \sqbrk {\mathfrak a} \subseteq \mathfrak a^e$
By Subset of Domain is Subset of Preimage of Image:
:$\mathfrak a \subseteq f^{-1} \sqbrk {f \sqbrk {\mathfrak a}}$
By Preimage of Subset is Subset of Preimage:
:$\mathfrak a \subseteq f^{-1} \sqbrk {f \sqbrk {\mathfrak a ... | Let $A$ and $B$ be [[Definition:Commutative Ring with Unity|commutative rings with unity]].
Let $f : A \to B$ be a [[Definition:unital Ring Homomorphism|ring homomorphism]].
Let $\mathfrak a \subseteq A$ be an [[Definition:Ideal of Ring|ideal]].
Then $\mathfrak a$ is [[Definition:Set Containment|contained]] in the ... | By definition of [[Definition:Extension of Ideal|extension]] and [[Definition:Generated Ideal of Ring|generated ideal]]:
:$f \sqbrk {\mathfrak a} \subseteq \mathfrak a^e$
By [[Subset of Domain is Subset of Preimage of Image]]:
:$\mathfrak a \subseteq f^{-1} \sqbrk {f \sqbrk {\mathfrak a}}$
By [[Preimage of Subset is ... | Ideal is Contained in Contraction of Extension | https://proofwiki.org/wiki/Ideal_is_Contained_in_Contraction_of_Extension | https://proofwiki.org/wiki/Ideal_is_Contained_in_Contraction_of_Extension | [
"Ideal Theory",
"Commutative Algebra",
"Ring Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:unital Ring Homomorphism",
"Definition:Ideal of Ring",
"Definition:Subset",
"Definition:Contraction of Ideal",
"Definition:Extension of Ideal"
] | [
"Definition:Extension of Ideal",
"Definition:Generated Ideal of Ring",
"Subset of Domain is Subset of Preimage of Image",
"Preimage of Subset is Subset of Preimage",
"Category:Ideal Theory",
"Category:Commutative Algebra",
"Category:Ring Theory"
] |
proofwiki-14179 | Equivalence of Definitions of Irreducible Polynomial over Field | Let $K$ be a field.
{{TFAE|def = Irreducible Polynomial}} | Note that by Field is Integral Domain, $K$ is indeed an integral domain.
{{proof wanted}}
Category:Irreducible Polynomials
e32op3s3m1i7mw9bk3haxpgjj5vsf1o | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
{{TFAE|def = Irreducible Polynomial}} | Note that by [[Field is Integral Domain]], $K$ is indeed an [[Definition:Integral Domain|integral domain]].
{{proof wanted}}
[[Category:Irreducible Polynomials]]
e32op3s3m1i7mw9bk3haxpgjj5vsf1o | Equivalence of Definitions of Irreducible Polynomial over Field | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Irreducible_Polynomial_over_Field | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Irreducible_Polynomial_over_Field | [
"Irreducible Polynomials"
] | [
"Definition:Field (Abstract Algebra)"
] | [
"Field is Integral Domain",
"Definition:Integral Domain",
"Category:Irreducible Polynomials"
] |
proofwiki-14180 | Linear Combination of Integrals/Indefinite | :$\ds \int \paren {\lambda \map f x + \mu \map g x} \rd x = \lambda \int \map f x \rd x + \mu \int \map g x \rd x$ | Let $F$ and $G$ be primitives of $f$ and $g$ respectively on $\closedint a b$.
By Linear Combination of Derivatives, $H = \lambda F + \mu G$ is a primitive of $\lambda f + \mu g$ on $\closedint a b$.
Hence:
{{begin-eqn}}
{{eqn | l = \int \paren {\lambda \map f t + \mu \map g t} \rd t
| r = \lambda \map F t + \mu ... | :$\ds \int \paren {\lambda \map f x + \mu \map g x} \rd x = \lambda \int \map f x \rd x + \mu \int \map g x \rd x$ | Let $F$ and $G$ be [[Definition:Primitive (Calculus)|primitives]] of $f$ and $g$ respectively on $\closedint a b$.
By [[Linear Combination of Derivatives]], $H = \lambda F + \mu G$ is a [[Definition:Primitive (Calculus)|primitive]] of $\lambda f + \mu g$ on $\closedint a b$.
Hence:
{{begin-eqn}}
{{eqn | l = \int \pa... | Linear Combination of Integrals/Indefinite | https://proofwiki.org/wiki/Linear_Combination_of_Integrals/Indefinite | https://proofwiki.org/wiki/Linear_Combination_of_Integrals/Indefinite | [
"Integral Calculus"
] | [] | [
"Definition:Primitive (Calculus)",
"Linear Combination of Derivatives",
"Definition:Primitive (Calculus)"
] |
proofwiki-14181 | Linear Combination of Integrals/Definite | :$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$ | Let $F$ and $G$ be primitives of $f$ and $g$ respectively on $\closedint a b$.
By Linear Combination of Derivatives, $H = \lambda F + \mu G$ is a primitive of $\lambda f + \mu g$ on $\closedint a b$.
Hence by the Fundamental Theorem of Calculus:
{{begin-eqn}}
{{eqn | l = \int_a^b \paren {\lambda \map f t + \mu \map g t... | :$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$ | Let $F$ and $G$ be [[Definition:Primitive (Calculus)|primitives]] of $f$ and $g$ respectively on $\closedint a b$.
By [[Linear Combination of Derivatives]], $H = \lambda F + \mu G$ is a [[Definition:Primitive (Calculus)|primitive]] of $\lambda f + \mu g$ on $\closedint a b$.
Hence by the [[Fundamental Theorem of Calc... | Linear Combination of Integrals/Definite/Proof 1 | https://proofwiki.org/wiki/Linear_Combination_of_Integrals/Definite | https://proofwiki.org/wiki/Linear_Combination_of_Integrals/Definite/Proof_1 | [
"Linear Combination of Definite Integrals",
"Definite Integrals"
] | [] | [
"Definition:Primitive (Calculus)",
"Linear Combination of Derivatives",
"Definition:Primitive (Calculus)",
"Fundamental Theorem of Calculus"
] |
proofwiki-14182 | Linear Combination of Integrals/Definite | :$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$ | It is clear that for step functions $s$ and $t$:
{{handwaving|"It is clear that ..."}}
{{MissingLinks}}
:$\ds \int_a^b \lambda \map s x + \mu \map t x \rd x = \lambda \int_a^b \map s x \rd x + \mu \int_a^b \map t x \rd x$
Under any partition, the lower Darboux sums and upper Darboux sums of $f$ and $g$ are step functio... | :$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$ | It is clear that for [[Definition:Step Function|step functions]] $s$ and $t$:
{{handwaving|"It is clear that ..."}}
{{MissingLinks}}
:$\ds \int_a^b \lambda \map s x + \mu \map t x \rd x = \lambda \int_a^b \map s x \rd x + \mu \int_a^b \map t x \rd x$
Under any partition, the [[Definition:Lower Darboux Sum|lower Darb... | Linear Combination of Integrals/Definite/Proof 2 | https://proofwiki.org/wiki/Linear_Combination_of_Integrals/Definite | https://proofwiki.org/wiki/Linear_Combination_of_Integrals/Definite/Proof_2 | [
"Linear Combination of Definite Integrals",
"Definite Integrals"
] | [] | [
"Definition:Step Function",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum",
"Definition:Step Function",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum",
"Definition:Lower Darboux Sum",
"Definition:Upper Darboux Sum",
"Definition:Lower Darboux Sum",
"Definition:Upper ... |
proofwiki-14183 | Correspondence between Abelian Groups and Z-Modules/Isomorphism of Categories | Let $\Z$ be the ring of integers.
Let $\mathbf{Ab}$ be the category of abelian groups.
Let $\mathbf{\mathbb Z-Mod}$ be the category of unitary $\Z$-modules.
Then the:
* forgetful functor $\mathbf{\mathbb Z-Mod} \to \mathbf{Ab}$
* associated Z-module functor $\mathbf{Ab} \to \mathbf{\mathbb Z-Mod}$
are strict inverse fu... | {{proof wanted}}
Category:Examples of Isomorphisms of Categories
Category:Category of Modules
Category:Category of Abelian Groups
tvzmplj60h6ilgkmjxv76mf8rkg4nml | Let $\Z$ be the [[Definition:Ring of Integers|ring of integers]].
Let $\mathbf{Ab}$ be the [[Definition:Category of Abelian Groups|category of abelian groups]].
Let $\mathbf{\mathbb Z-Mod}$ be the [[Definition:Category of Unitary Modules|category of unitary]] $\Z$-modules.
Then the:
* [[Definition:Forgetful Functor... | {{proof wanted}}
[[Category:Examples of Isomorphisms of Categories]]
[[Category:Category of Modules]]
[[Category:Category of Abelian Groups]]
tvzmplj60h6ilgkmjxv76mf8rkg4nml | Correspondence between Abelian Groups and Z-Modules/Isomorphism of Categories | https://proofwiki.org/wiki/Correspondence_between_Abelian_Groups_and_Z-Modules/Isomorphism_of_Categories | https://proofwiki.org/wiki/Correspondence_between_Abelian_Groups_and_Z-Modules/Isomorphism_of_Categories | [
"Examples of Isomorphisms of Categories",
"Category of Modules",
"Category of Abelian Groups"
] | [
"Definition:Ring of Integers",
"Definition:Category of Abelian Groups",
"Definition:Category of Unitary Modules",
"Definition:Forgetful Functor from Modules to Abelian Groups",
"Definition:Associated Z-Module Functor",
"Definition:Strict Inverse Functors",
"Definition:Isomorphism of Categories/Isomorphi... | [
"Category:Examples of Isomorphisms of Categories",
"Category:Category of Modules",
"Category:Category of Abelian Groups"
] |
proofwiki-14184 | Correspondence between Abelian Groups and Z-Modules/Homomorphisms | Let $G, H$ be abelian groups.
Let $f : G \to H$ be a mapping.
{{TFAE}}
:$(1): \quad f$ is a group homomorphism.
:$(2): \quad f$ is a $\Z$-module homomorphism between the $\Z$-modules associated with $G$ and $H$. | {{proof wanted}}
Category:Z-Module Associated with Abelian Group
Category:Abelian Groups
Category:Group Homomorphisms
Category:Linear Transformations
6tpem2e4d6yhy6iek9epmlpiw2q2d3t | Let $G, H$ be [[Definition:Abelian Group|abelian groups]].
Let $f : G \to H$ be a [[Definition:mapping|mapping]].
{{TFAE}}
:$(1): \quad f$ is a [[Definition:Group Homomorphism|group homomorphism]].
:$(2): \quad f$ is a [[Definition:Module Homomorphism|$\Z$-module homomorphism]] between the [[Definition:Z-Module Asso... | {{proof wanted}}
[[Category:Z-Module Associated with Abelian Group]]
[[Category:Abelian Groups]]
[[Category:Group Homomorphisms]]
[[Category:Linear Transformations]]
6tpem2e4d6yhy6iek9epmlpiw2q2d3t | Correspondence between Abelian Groups and Z-Modules/Homomorphisms | https://proofwiki.org/wiki/Correspondence_between_Abelian_Groups_and_Z-Modules/Homomorphisms | https://proofwiki.org/wiki/Correspondence_between_Abelian_Groups_and_Z-Modules/Homomorphisms | [
"Z-Module Associated with Abelian Group",
"Abelian Groups",
"Group Homomorphisms",
"Linear Transformations"
] | [
"Definition:Abelian Group",
"Definition:mapping",
"Definition:Group Homomorphism",
"Definition:Linear Transformation",
"Definition:Z-Module Associated with Abelian Group"
] | [
"Category:Z-Module Associated with Abelian Group",
"Category:Abelian Groups",
"Category:Group Homomorphisms",
"Category:Linear Transformations"
] |
proofwiki-14185 | Definite Integral of Constant Multiple of Real Function | Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.
{{mistake|$f$ is only integrable here, but theorems used requiring that $f$ is continuous are used in both proofs}}
Let $c \in \R$ be a real number.
Then:
:$\ds \int_a^b c \map f x \rd x = c \int_a^b \map f x \rd x$ | Let $F$ be a primitive of $f$ on $\closedint a b$.
By Primitive of Constant Multiple of Function, $H = c F$ is a primitive of $c f$ on $\closedint a b$.
Hence by the Fundamental Theorem of Calculus:
{{begin-eqn}}
{{eqn | l = \int_a^b c \map f x \rd x
| r = \bigintlimits {c \map F x} a b
| c =
}}
{{eqn | r ... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
{{mistake|$f$ is only integrable here, but theorems used requiring that $f$ is continuous are used in both proofs}}
Let $c \in \R$ ... | Let $F$ be a [[Definition:Primitive (Calculus)|primitive]] of $f$ on $\closedint a b$.
By [[Primitive of Constant Multiple of Function]], $H = c F$ is a [[Definition:Primitive (Calculus)|primitive]] of $c f$ on $\closedint a b$.
Hence by the [[Fundamental Theorem of Calculus]]:
{{begin-eqn}}
{{eqn | l = \int_a^b c \... | Definite Integral of Constant Multiple of Real Function/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_of_Constant_Multiple_of_Real_Function | https://proofwiki.org/wiki/Definite_Integral_of_Constant_Multiple_of_Real_Function/Proof_1 | [
"Definite Integral of Constant Multiple of Real Function",
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Real Number"
] | [
"Definition:Primitive (Calculus)",
"Primitive of Constant Multiple of Function",
"Definition:Primitive (Calculus)",
"Fundamental Theorem of Calculus"
] |
proofwiki-14186 | Definite Integral of Constant Multiple of Real Function | Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.
{{mistake|$f$ is only integrable here, but theorems used requiring that $f$ is continuous are used in both proofs}}
Let $c \in \R$ be a real number.
Then:
:$\ds \int_a^b c \map f x \rd x = c \int_a^b \map f x \rd x$ | Let $F$ be a primitive of $f$ on $\closedint a b$.
By Linear Combination of Definite Integrals:
:$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$
for real functions $f$ and $g$ which are integrable on the closed interval $\closedint a b$, wher... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
{{mistake|$f$ is only integrable here, but theorems used requiring that $f$ is continuous are used in both proofs}}
Let $c \in \R$ ... | Let $F$ be a [[Definition:Primitive (Calculus)|primitive]] of $f$ on $\closedint a b$.
By [[Linear Combination of Definite Integrals]]:
:$\ds \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$
for [[Definition:Real Function|real functions]] $f$ an... | Definite Integral of Constant Multiple of Real Function/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_of_Constant_Multiple_of_Real_Function | https://proofwiki.org/wiki/Definite_Integral_of_Constant_Multiple_of_Real_Function/Proof_2 | [
"Definite Integral of Constant Multiple of Real Function",
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Real Number"
] | [
"Definition:Primitive (Calculus)",
"Linear Combination of Integrals/Definite",
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Real Number"
] |
proofwiki-14187 | Mapping to Singleton is Unique | Let $S$ be a set.
Let $T$ be a singleton.
Then there exists a unique mapping $S \to T$. | Let $T = \set t$.
Let $f$ and $g$ both be mappings from $S$ to $T$.
From Mapping is Constant iff Image is Singleton:
:$\forall s \in S: \map f s = t$
and:
:$\forall s \in S: \map g s = t$
The result follows by Equality of Mappings.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $T$ be a [[Definition:Singleton|singleton]].
Then there exists a [[Definition:Unique|unique]] [[Definition:Mapping|mapping]] $S \to T$. | Let $T = \set t$.
Let $f$ and $g$ both be [[Definition:Mapping|mappings]] from $S$ to $T$.
From [[Mapping is Constant iff Image is Singleton]]:
:$\forall s \in S: \map f s = t$
and:
:$\forall s \in S: \map g s = t$
The result follows by [[Equality of Mappings]].
{{qed}} | Mapping to Singleton is Unique | https://proofwiki.org/wiki/Mapping_to_Singleton_is_Unique | https://proofwiki.org/wiki/Mapping_to_Singleton_is_Unique | [
"Singletons"
] | [
"Definition:Set",
"Definition:Singleton",
"Definition:Unique",
"Definition:Mapping"
] | [
"Definition:Mapping",
"Mapping is Constant iff Image is Singleton",
"Equality of Mappings"
] |
proofwiki-14188 | Mean Value Theorem for Integrals/Generalization | Let $f$ and $g$ be continuous real functions on the closed interval $\closedint a b$ such that:
:$\forall x \in \closedint a b: \map g x \ge 0$
Then there exists a real number $k \in \closedint a b$ such that:
:$\ds \int_a^b \map f x \map g x \rd x = \map f k \int_a^b \map g x \rd x$ | Let:
:$\ds \int_a^b \map g x \rd x = 0$
We are given that:
:$\forall x \in \closedint a b: \map g x \ge 0$
Hence by Continuous Non-Negative Real Function with Zero Integral is Zero Function:
:$\forall x \in \closedint a b: \map g x = 0$
Hence:
:$\ds \int_a^b \map f x \cdot 0 \rd x = \map f k \cdot 0$
and so the result ... | Let $f$ and $g$ be [[Definition:Continuous Real Function on Closed Interval|continuous real functions]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$ such that:
:$\forall x \in \closedint a b: \map g x \ge 0$
Then there exists a [[Definition:Real Number|real number]] $k \in \closedint a b... | Let:
:$\ds \int_a^b \map g x \rd x = 0$
We are given that:
:$\forall x \in \closedint a b: \map g x \ge 0$
Hence by [[Continuous Non-Negative Real Function with Zero Integral is Zero Function]]:
:$\forall x \in \closedint a b: \map g x = 0$
Hence:
:$\ds \int_a^b \map f x \cdot 0 \rd x = \map f k \cdot 0$
and so t... | Mean Value Theorem for Integrals/Generalization | https://proofwiki.org/wiki/Mean_Value_Theorem_for_Integrals/Generalization | https://proofwiki.org/wiki/Mean_Value_Theorem_for_Integrals/Generalization | [
"Definite Integrals",
"Named Theorems"
] | [
"Definition:Continuous Real Function/Closed Interval",
"Definition:Real Interval/Closed",
"Definition:Real Number"
] | [
"Continuous Non-Negative Real Function with Zero Integral is Zero Function",
"Continuous Real Function is Darboux Integrable",
"Definition:Darboux Integrable Function",
"Extreme Value Theorem",
"Definition:Primitive (Calculus)/Integration",
"Linear Combination of Integrals/Definite",
"Intermediate Value... |
proofwiki-14189 | Rectangular Formula for Definite Integrals | Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.
Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$:
:$\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$
Then the definite integral of $f$ {{WRT|Integration}} ... | {{ProofWanted|This will probably boil down to a graphical approach based on the structure of a Darboux integral. We need to explain rigorously what an "approximation" means, and we may also want to quantify the error.}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a [[Definition:Normal Subdivision|normal subdivision]] of $\closedint a... | {{ProofWanted|This will probably boil down to a graphical approach based on the structure of a [[Definition:Darboux Integral|Darboux integral]]. We need to explain rigorously what an "approximation" means, and we may also want to quantify the error.}} | Rectangular Formula for Definite Integrals | https://proofwiki.org/wiki/Rectangular_Formula_for_Definite_Integrals | https://proofwiki.org/wiki/Rectangular_Formula_for_Definite_Integrals | [
"Definite Integrals"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Subdivision of Interval/Normal Subdivision",
"Definition:Definite Integral"
] | [
"Definition:Definite Integral/Darboux"
] |
proofwiki-14190 | Trapezium Rule for Definite Integrals | Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.
Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$:
:$\forall i \in \set {1, 2, \ldots, n}: x_i - x_{i - 1} = \dfrac {b - a} n$
Then the definite integral of $f$ {{WRT|Integration}} ... | The geometric interpretation of a definite integral states that the area between the $4$ lines $x = a$, $x = b$, $y = 0$ and $y = \map f x$ is equal to $\ds \int_a^b \map f x \rd x$.
We approximate this area by dividing it into trapezia:
:460px
Consider the trapezium $T_i$ whose vertices are $\tuple {x_i, \map f {x_i},... | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a [[Definition:Normal Subdivision|normal subdivision]] of $\closedint a... | The [[Definition:Darboux Integral/Geometric Interpretation|geometric interpretation]] of a [[Definition:Definite Integral|definite integral]] states that the [[Definition:Area|area]] between the $4$ [[Definition:Line|lines]] $x = a$, $x = b$, $y = 0$ and $y = \map f x$ is equal to $\ds \int_a^b \map f x \rd x$.
We [[... | Trapezium Rule for Definite Integrals | https://proofwiki.org/wiki/Trapezium_Rule_for_Definite_Integrals | https://proofwiki.org/wiki/Trapezium_Rule_for_Definite_Integrals | [
"Trapezium Rule for Definite Integrals",
"Newton-Cotes Rules",
"Definite Integrals",
"Trapezia"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Definition:Subdivision of Interval/Normal Subdivision",
"Definition:Definite Integral",
"Definition:Approximation"
] | [
"Definition:Darboux Integral/Geometric Interpretation",
"Definition:Definite Integral",
"Definition:Area",
"Definition:Line",
"Definition:Approximation",
"Definition:Area",
"Definition:Quadrilateral/Trapezium",
"File:Trapezium-rule.png",
"Definition:Quadrilateral/Trapezium",
"Definition:Polygon/Ve... |
proofwiki-14191 | Simpson's Rule | Let $f$ be a real function which is integrable on the closed interval $\closedint a b$.
:$\ds \int_a^b \map f x \rd x \approx \dfrac {b - a} 6 \paren {\map f a + 4 \map f {\dfrac {a + b} 2} + \map f b}$
Hence the area under the curve is approximated by the area under the quadratic polynomial passing through $\tuple {x,... | {{ProofWanted|Graphical approach based on approximating the area under the curve as a series of parabolas.}} | Let $f$ be a [[Definition:Real Function|real function]] which is [[Definition:Integrable Function|integrable]] on the [[Definition:Closed Real Interval|closed interval]] $\closedint a b$.
:$\ds \int_a^b \map f x \rd x \approx \dfrac {b - a} 6 \paren {\map f a + 4 \map f {\dfrac {a + b} 2} + \map f b}$
Hence the [[Ar... | {{ProofWanted|Graphical approach based on approximating the area under the curve as a series of parabolas.}} | Simpson's Rule | https://proofwiki.org/wiki/Simpson's_Rule | https://proofwiki.org/wiki/Simpson's_Rule | [
"Simpson's Rule",
"Definite Integrals",
"Numerical Integration"
] | [
"Definition:Real Function",
"Definition:Integrable Function",
"Definition:Real Interval/Closed",
"Area under Curve",
"Area under Curve",
"Definition:Quadratic Polynomial",
"Simpson's Rule/Repeated"
] | [] |
proofwiki-14192 | Definite Integral to Infinity of Reciprocal of x Squared plus a Squared | :$\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$
for $a \ne 0$. | {{begin-eqn}}
{{eqn | l = \int_0^\infty \dfrac {\d x} {x^2 + a^2}
| r = \int_0^{\mathop \to +\infty} \dfrac {\d x} {x^2 + a^2}
| c =
}}
{{eqn | r = \lim_{\gamma \mathop \to +\infty} \int_0^\gamma \dfrac {\d x} {x^2 + a^2}
| c = {{Defof|Improper Integral on Closed Interval Unbounded Above}}
}}
{{eqn |... | :$\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$
for $a \ne 0$. | {{begin-eqn}}
{{eqn | l = \int_0^\infty \dfrac {\d x} {x^2 + a^2}
| r = \int_0^{\mathop \to +\infty} \dfrac {\d x} {x^2 + a^2}
| c =
}}
{{eqn | r = \lim_{\gamma \mathop \to +\infty} \int_0^\gamma \dfrac {\d x} {x^2 + a^2}
| c = {{Defof|Improper Integral on Closed Interval Unbounded Above}}
}}
{{eqn |... | Definite Integral to Infinity of Reciprocal of x Squared plus a Squared/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_x_Squared_plus_a_Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_x_Squared_plus_a_Squared/Proof_1 | [
"Definite Integral to Infinity of Reciprocal of x Squared plus a Squared",
"Definite Integrals involving x squared plus a squared"
] | [] | [
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Arctangent of Zero is Zero",
"Arctangent Tends to Half Pi as Argument Tends to Infinity"
] |
proofwiki-14193 | Definite Integral to Infinity of Reciprocal of x Squared plus a Squared | :$\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$
for $a \ne 0$. | {{begin-eqn}}
{{eqn | l = \int_0^\infty \dfrac {\d x} {x^2 + a^2}
| r = \frac \pi { 2 a^{2 - 1} } \csc \left({\frac \pi 2}\right)
| c = Definite Integral to Infinity of $\dfrac 1 {1 + x^n}$: Corollary
}}
{{eqn | r = \frac \pi {2 a}
| c = Sine of Right Angle
}}
{{end-eqn}}
{{qed}} | :$\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$
for $a \ne 0$. | {{begin-eqn}}
{{eqn | l = \int_0^\infty \dfrac {\d x} {x^2 + a^2}
| r = \frac \pi { 2 a^{2 - 1} } \csc \left({\frac \pi 2}\right)
| c = [[Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Corollary|Definite Integral to Infinity of $\dfrac 1 {1 + x^n}$: Corollary]]
}}
{{eqn | r = \frac \pi {2 a}
| c =... | Definite Integral to Infinity of Reciprocal of x Squared plus a Squared/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_x_Squared_plus_a_Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_x_Squared_plus_a_Squared/Proof_2 | [
"Definite Integral to Infinity of Reciprocal of x Squared plus a Squared",
"Definite Integrals involving x squared plus a squared"
] | [] | [
"Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Corollary",
"Sine of Right Angle"
] |
proofwiki-14194 | Definite Integral to Infinity of Reciprocal of x Squared plus a Squared | :$\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$
for $a \ne 0$. | Let $C_R$ be the semicircular contour of radius $R$ situated on the upper half plane, centred at the origin, traversed anti-clockwise.
Let $\Gamma_R = C_R \cup \closedint {-R} R$.
Then, by Contour Integral of Concatenation of Contours:
:$\ds \oint_{\Gamma_R} \frac {\d z} {z^2 + a^2} = \int_{C_R} \frac {\d z} {z^2 + a^2... | :$\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$
for $a \ne 0$. | Let $C_R$ be the semicircular [[Definition:Contour (Complex Plane)|contour]] of radius $R$ situated on the upper [[Definition:Half-Plane|half plane]], centred at the origin, traversed anti-clockwise.
Let $\Gamma_R = C_R \cup \closedint {-R} R$.
Then, by [[Contour Integral of Concatenation of Contours]]:
:$\ds \oint_... | Definite Integral to Infinity of Reciprocal of x Squared plus a Squared/Proof 3 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_x_Squared_plus_a_Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_x_Squared_plus_a_Squared/Proof_3 | [
"Definite Integral to Infinity of Reciprocal of x Squared plus a Squared",
"Definite Integrals involving x squared plus a squared"
] | [] | [
"Definition:Contour/Complex Plane",
"Definition:Half-Plane",
"Contour Integral of Concatenation of Contours",
"Estimation Lemma for Contour Integrals",
"Definition:Integration/Integrand",
"Definition:Meromorphic Function",
"Cauchy's Residue Theorem",
"Definition:Summation",
"Definition:Isolated Sing... |
proofwiki-14195 | Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Corollary | :$\ds \int_0^\infty \frac 1 {a^n + x^n} \rd x = \frac \pi {n a^{n - 1} } \map \csc {\frac \pi n}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac 1 {a^n + x^n} \rd x
| r = \frac 1 {a^n} \int_0^\infty \frac 1 {1 + \paren {\frac x a}^n} \rd x
}}
{{eqn | r = \frac 1 {a^n} \cdot \frac 1 {\frac 1 a} \int_0^\infty \frac 1 {1 + \paren {\frac x a}^n} \map \rd {\frac x a}
| c = Primitive of Function of Constant Multiple
}}
... | :$\ds \int_0^\infty \frac 1 {a^n + x^n} \rd x = \frac \pi {n a^{n - 1} } \map \csc {\frac \pi n}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac 1 {a^n + x^n} \rd x
| r = \frac 1 {a^n} \int_0^\infty \frac 1 {1 + \paren {\frac x a}^n} \rd x
}}
{{eqn | r = \frac 1 {a^n} \cdot \frac 1 {\frac 1 a} \int_0^\infty \frac 1 {1 + \paren {\frac x a}^n} \map \rd {\frac x a}
| c = [[Primitive of Function of Constant Multiple]]... | Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Corollary | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_1_plus_Power_of_x/Corollary | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_1_plus_Power_of_x/Corollary | [
"Definite Integral to Infinity of Reciprocal of 1 plus Power of x"
] | [] | [
"Primitive of Function of Constant Multiple",
"Definite Integral to Infinity of Reciprocal of 1 plus Power of x",
"Category:Definite Integral to Infinity of Reciprocal of 1 plus Power of x"
] |
proofwiki-14196 | Definite Integral to Infinity of Power of x over 1 + x | :$\ds \int_0^\infty \dfrac {x^{p - 1} \rd x} {1 + x} = \frac \pi {\sin \pi p}$
for $0 < p < 1$. | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {x^{p - 1} \rd x} {1 + x}
| r = \int_0^\infty \frac 1 {p x^{p - 1} } \cdot \frac {x^{p - 1} \rd t} {1 + t^{1 / p} }
| c = substituting $t = x^p$
}}
{{eqn | r = \frac 1 p \int_0^\infty \frac {\rd t} {1 + t^{1 / p} }
}}
{{eqn | r = \frac 1 p \cdot \frac \pi {\frac 1 p} \ma... | :$\ds \int_0^\infty \dfrac {x^{p - 1} \rd x} {1 + x} = \frac \pi {\sin \pi p}$
for $0 < p < 1$. | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {x^{p - 1} \rd x} {1 + x}
| r = \int_0^\infty \frac 1 {p x^{p - 1} } \cdot \frac {x^{p - 1} \rd t} {1 + t^{1 / p} }
| c = [[Integration by Substitution|substituting]] $t = x^p$
}}
{{eqn | r = \frac 1 p \int_0^\infty \frac {\rd t} {1 + t^{1 / p} }
}}
{{eqn | r = \frac 1 p... | Definite Integral to Infinity of Power of x over 1 + x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_over_1_+_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_over_1_+_x/Proof_1 | [
"Definite Integral to Infinity of Power of x over 1 + x",
"Examples of Definite Integrals"
] | [] | [
"Integration by Substitution",
"Definite Integral to Infinity of Reciprocal of 1 plus Power of x"
] |
proofwiki-14197 | Definite Integral to Infinity of Power of x over 1 + x | :$\ds \int_0^\infty \dfrac {x^{p - 1} \rd x} {1 + x} = \frac \pi {\sin \pi p}$
for $0 < p < 1$. | {{ProofWanted|We can probably use Primitive of Power of x by Power of a x + b but I can see it being a long slog}} | :$\ds \int_0^\infty \dfrac {x^{p - 1} \rd x} {1 + x} = \frac \pi {\sin \pi p}$
for $0 < p < 1$. | {{ProofWanted|We can probably use [[Primitive of Power of x by Power of a x + b]] but I can see it being a long slog}} | Definite Integral to Infinity of Power of x over 1 + x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_over_1_+_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_over_1_+_x/Proof_2 | [
"Definite Integral to Infinity of Power of x over 1 + x",
"Examples of Definite Integrals"
] | [] | [
"Primitive of Power of x by Power of a x + b"
] |
proofwiki-14198 | Definite Integral to Infinity of Power of x over Power of x plus Power of a | :$\ds \int_0^\infty \dfrac {x^m \rd x} {x^n + a^n} = \frac {\pi a^{m + 1 - n} } {n \map \sin {\paren {m + 1} \frac \pi n} }$
for $0 < m + 1 < n$. | {{begin-eqn}}
{{eqn | l = \int_0^\infty \dfrac {x^m \rd x} {x^n + a^n}
| r = \int_0^\infty \dfrac {x^m \rd x} {\paren {x^{m + 1} }^{\frac n {m + 1} } + \paren {a^{m + 1} }^{\frac n {m + 1} } }
}}
{{eqn | r = \frac 1 {m + 1} \int_0^\infty \dfrac 1 {u^{\frac n {m + 1} } + \paren {a^{m + 1} }^{\frac n {m + 1} } } \r... | :$\ds \int_0^\infty \dfrac {x^m \rd x} {x^n + a^n} = \frac {\pi a^{m + 1 - n} } {n \map \sin {\paren {m + 1} \frac \pi n} }$
for $0 < m + 1 < n$. | {{begin-eqn}}
{{eqn | l = \int_0^\infty \dfrac {x^m \rd x} {x^n + a^n}
| r = \int_0^\infty \dfrac {x^m \rd x} {\paren {x^{m + 1} }^{\frac n {m + 1} } + \paren {a^{m + 1} }^{\frac n {m + 1} } }
}}
{{eqn | r = \frac 1 {m + 1} \int_0^\infty \dfrac 1 {u^{\frac n {m + 1} } + \paren {a^{m + 1} }^{\frac n {m + 1} } } \r... | Definite Integral to Infinity of Power of x over Power of x plus Power of a | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_over_Power_of_x_plus_Power_of_a | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_over_Power_of_x_plus_Power_of_a | [
"Examples of Definite Integrals"
] | [] | [
"Integration by Substitution",
"Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Corollary",
"Cosecant is Reciprocal of Sine"
] |
proofwiki-14199 | Definite Integral to Infinity of Power of x over 1 + 2 x Cosine Beta + x Squared | :$\ds \int_0^\infty \dfrac {x^m \rd x} {1 + 2 x \cos \beta + x^2} = \frac \pi {\sin m x} \frac {\sin m \beta} {\sin \beta}$ | {{ProofWanted|write $x^2 + 2 x \cos \beta + 1 \equiv \paren {x + e^{i \beta} } \paren {x + e^{-i \beta} }$ then use partial fractions. Ugly so will do later.}} | :$\ds \int_0^\infty \dfrac {x^m \rd x} {1 + 2 x \cos \beta + x^2} = \frac \pi {\sin m x} \frac {\sin m \beta} {\sin \beta}$ | {{ProofWanted|write $x^2 + 2 x \cos \beta + 1 \equiv \paren {x + e^{i \beta} } \paren {x + e^{-i \beta} }$ then use partial fractions. Ugly so will do later.}} | Definite Integral to Infinity of Power of x over 1 + 2 x Cosine Beta + x Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_over_1_+_2_x_Cosine_Beta_+_x_Squared | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_over_1_+_2_x_Cosine_Beta_+_x_Squared | [
"Examples of Definite Integrals"
] | [] | [] |
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