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-16200 | Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 3 | Let $M = \struct {A, d}$ be a metric space or a pseudometric space.
Let $\sequence {x_k}$ be a sequence in $A$.
{{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}} | By definition of a convergent real sequence:
:$\ds \lim_{n \mathop \to \infty} \map d {x_n, l} = 0$
{{iff}}
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {\map d {x_n, l} - 0} < \epsilon$
From Distance in Pseudometric is Non-Negative:
:$\forall x, y \in A: \map d {x, y} \ge 0$
Hence:
:$\f... | Let $M = \struct {A, d}$ be a [[Definition:Metric Space|metric space]] or a [[Definition:Pseudometric Space|pseudometric space]].
Let $\sequence {x_k}$ be a [[Definition:Sequence|sequence in $A$]].
{{TFAE|def = Convergent Sequence|context = Metric Space|contextview = Metric Spaces}} | By definition of a [[Definition:Convergent Real Sequence|convergent real sequence]]:
:$\ds \lim_{n \mathop \to \infty} \map d {x_n, l} = 0$
{{iff}}
:$\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {\map d {x_n, l} - 0} < \epsilon$
From [[Distance in Pseudometric is Non-Negative]]:
:$\foral... | Equivalence of Definitions of Convergent Sequence in Metric Space/Definition 1 iff Definition 3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_Sequence_in_Metric_Space/Definition_1_iff_Definition_3 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Convergent_Sequence_in_Metric_Space/Definition_1_iff_Definition_3 | [
"Equivalence of Definitions of Convergent Sequence in Metric Space"
] | [
"Definition:Metric Space",
"Definition:Pseudometric/Pseudometric Space",
"Definition:Sequence"
] | [
"Definition:Convergent Sequence/Real Numbers",
"Distance in Pseudometric is Non-Negative"
] |
proofwiki-16201 | Definite Integral from 0 to 1 of Logarithm of x over One minus x | :$\ds \int_0^1 \frac {\ln x} {1 - x} \rd x = -\frac {\pi^2} 6$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\ln x} {1 - x} \rd x
| r = \int_0^1 \ln x \paren {\sum_{n \mathop = 0}^\infty x^n} \rd x
| c = Sum of Infinite Geometric Sequence
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {\int_0^1 x^n \ln x \rd x}
| c = Fubini's Theorem
}}
{{eqn | r = -\sum_{n \matho... | :$\ds \int_0^1 \frac {\ln x} {1 - x} \rd x = -\frac {\pi^2} 6$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\ln x} {1 - x} \rd x
| r = \int_0^1 \ln x \paren {\sum_{n \mathop = 0}^\infty x^n} \rd x
| c = [[Sum of Infinite Geometric Sequence]]
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren {\int_0^1 x^n \ln x \rd x}
| c = [[Fubini's Theorem]]
}}
{{eqn | r = -\sum_{... | Definite Integral from 0 to 1 of Logarithm of x over One minus x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_over_One_minus_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_over_One_minus_x | [
"Definite Integrals involving Logarithm Function"
] | [] | [
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x",
"Gamma Function Extends Factorial",
"Basel Problem"
] |
proofwiki-16202 | Definite Integral from 0 to 1 of Logarithm of One plus x over x | :$\ds \int_0^1 \frac {\map \ln {1 + x} } x \rd x = \frac {\pi^2} {12}$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\map \ln {1 + x} } x \rd x
| r = \int_0^1 \frac 1 x \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n} \rd x
| c = Power Series Expansion for $\map \ln {1 + x}$
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \paren {\frac {\paren {-1}^{n - 1} } n \int_0^1 x^{n... | :$\ds \int_0^1 \frac {\map \ln {1 + x} } x \rd x = \frac {\pi^2} {12}$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\map \ln {1 + x} } x \rd x
| r = \int_0^1 \frac 1 x \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n} \rd x
| c = [[Power Series Expansion for Logarithm of 1 + x|Power Series Expansion for $\map \ln {1 + x}$]]
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \p... | Definite Integral from 0 to 1 of Logarithm of One plus x over x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_One_plus_x_over_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_One_plus_x_over_x | [
"Definite Integrals involving Logarithm Function"
] | [] | [
"Power Series Expansion for Logarithm of 1 + x",
"Power Series is Termwise Integrable within Radius of Convergence",
"Primitive of Power",
"Sum of Reciprocals of Squares Alternating in Sign"
] |
proofwiki-16203 | Definite Integral from 0 to 1 of Logarithm of One minus x over x | :$\ds \int_0^1 \frac {\map \ln {1 - x} } x \rd x = -\frac {\pi^2} 6$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\map \ln {1 - x} } x \rd x
| r = -\int_0^1 \frac 1 x \paren {\sum_{n \mathop = 1}^\infty \frac {x^n} n} \rd x
| c = Power Series Expansion for $\map \ln {1 - x}$
}}
{{eqn | r = -\sum_{n \mathop = 1}^\infty \paren {\frac 1 n \int_0^1 x^{n - 1} \rd x}
| c = Pow... | :$\ds \int_0^1 \frac {\map \ln {1 - x} } x \rd x = -\frac {\pi^2} 6$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\map \ln {1 - x} } x \rd x
| r = -\int_0^1 \frac 1 x \paren {\sum_{n \mathop = 1}^\infty \frac {x^n} n} \rd x
| c = [[Power Series Expansion for Logarithm of 1 - x|Power Series Expansion for $\map \ln {1 - x}$]]
}}
{{eqn | r = -\sum_{n \mathop = 1}^\infty \paren {\f... | Definite Integral from 0 to 1 of Logarithm of One minus x over x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_One_minus_x_over_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_One_minus_x_over_x | [
"Definite Integrals involving Logarithm Function"
] | [] | [
"Power Series Expansion for Logarithm of 1 - x",
"Power Series is Termwise Integrable within Radius of Convergence",
"Primitive of Power",
"Basel Problem"
] |
proofwiki-16204 | Definite Integral to Infinity of Exponential of -a x by Cosine of b x | :$\ds \int_0^\infty e^{-a x} \cos b x \rd x = \frac a {a^2 + b^2}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty e^{-a x} \cos b x \rd x
| r = \intlimits {\frac {e^{-a x} \paren {-a \cos b x + b \sin b x} } {a^2 + b^2} } 0 \infty
| c = Primitive of $e^{a x} \cos b x$
}}
{{eqn | r = \lim_{x \mathop \to \infty} \paren {\frac {e^{-a x} \paren {-a \cos b x + b \sin b x} } {a^2 + b^2} } + \fra... | :$\ds \int_0^\infty e^{-a x} \cos b x \rd x = \frac a {a^2 + b^2}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty e^{-a x} \cos b x \rd x
| r = \intlimits {\frac {e^{-a x} \paren {-a \cos b x + b \sin b x} } {a^2 + b^2} } 0 \infty
| c = [[Primitive of Exponential of a x by Cosine of b x|Primitive of $e^{a x} \cos b x$]]
}}
{{eqn | r = \lim_{x \mathop \to \infty} \paren {\frac {e^{-a x} \pa... | Definite Integral to Infinity of Exponential of -a x by Cosine of b x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_by_Cosine_of_b_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_by_Cosine_of_b_x | [
"Definite Integrals involving Cosine Function",
"Definite Integrals involving Exponential Function"
] | [] | [
"Primitive of Exponential of a x by Cosine of b x",
"Exponential of Zero",
"Cosine of Zero is One",
"Sine of Zero is Zero",
"Linear Combination of Sine and Cosine",
"Exponential Tends to Zero and Infinity",
"Squeeze Theorem"
] |
proofwiki-16205 | Set of Sets can be Defined as Family | Let $\Bbb S$ be a set of sets.
Then $\Bbb S$ can be defined as an indexed family of sets. | Let $S: \Bbb S \to \Bbb S$ denote the identity mapping on $\Bbb S$:
:$\forall i \in \Bbb S: S_i = i$
where we use $S_i$ to mean the image of $i$ under $S$:
:$S_i := \map S i$
Then we can consider $S$ as an indexing function from $\Bbb S$ to $\Bbb S$.
Hence in this case $\Bbb S$ is at the same time both:
:an indexing se... | Let $\Bbb S$ be a [[Definition:Set of Sets|set of sets]].
Then $\Bbb S$ can be defined as an [[Definition:Indexed Family of Sets|indexed family of sets]]. | Let $S: \Bbb S \to \Bbb S$ denote the [[Definition:Identity Mapping|identity mapping]] on $\Bbb S$:
:$\forall i \in \Bbb S: S_i = i$
where we use $S_i$ to mean the [[Definition:Image of Element under Mapping|image]] of $i$ under $S$:
:$S_i := \map S i$
Then we can consider $S$ as an [[Definition:Indexing Function|ind... | Set of Sets can be Defined as Family | https://proofwiki.org/wiki/Set_of_Sets_can_be_Defined_as_Family | https://proofwiki.org/wiki/Set_of_Sets_can_be_Defined_as_Family | [
"Set Systems",
"Indexed Families"
] | [
"Definition:Set of Sets",
"Definition:Indexing Set/Family of Sets"
] | [
"Definition:Identity Mapping",
"Definition:Image (Set Theory)/Mapping/Element",
"Definition:Indexing Set/Function",
"Definition:Indexing Set",
"Definition:Indexing Set/Indexed Set",
"Definition:Set",
"Definition:Indexing Set/Index",
"Definition:Indexing Set/Term",
"Definition:Indexing Set/Family"
] |
proofwiki-16206 | Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x | :$\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } x \rd x = \ln \frac b a$ | Note that the integrand is of the form:
:$\ds \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x$
where:
:$\map f x = e^{-x}$
We have, by Derivative of Exponential Function:
:$\map {f'} x = -e^{-x}$
which is continuous on $\R$.
We also have, by Exponential Tends to Zero and Infinity:
:$\ds \lim_{x \mathop \to... | :$\ds \int_0^\infty \frac {e^{-a x} - e^{-b x} } x \rd x = \ln \frac b a$ | Note that the [[Definition:Integrand|integrand]] is of the form:
:$\ds \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x$
where:
:$\map f x = e^{-x}$
We have, by [[Derivative of Exponential Function]]:
:$\map {f'} x = -e^{-x}$
which is [[Definition:Continuous Real Function|continuous]] on $\R$.
We also... | Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_minus_Exponential_of_-b_x_over_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x_minus_Exponential_of_-b_x_over_x | [
"Definite Integrals involving Exponential Function"
] | [] | [
"Definition:Integration/Integrand",
"Derivative of Exponential Function",
"Definition:Continuous Real Function",
"Exponential Tends to Zero and Infinity",
"Definition:Continuously Differentiable",
"Frullani's Integral",
"Exponential of Zero",
"Logarithm of Reciprocal"
] |
proofwiki-16207 | Definite Integral to Infinity of Exponential of -a x^2 | :$\ds \int_0^\infty e^{-a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi a}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty e^{-a x^2} \rd x
| r = \int_0^\infty e^{-\paren {\sqrt a x}^2} \rd x
}}
{{eqn | r = \frac 1 {\sqrt a} \int_0^\infty e^{-t^2} \rd t
| c = substituting $t = \sqrt a x$
}}
{{eqn | r = \frac 1 2 \sqrt {\frac \pi a}
| c = Integral to Infinity of $e^{-t^2}$
}}
{{end-eqn}}
{{qed}} | :$\ds \int_0^\infty e^{-a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi a}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty e^{-a x^2} \rd x
| r = \int_0^\infty e^{-\paren {\sqrt a x}^2} \rd x
}}
{{eqn | r = \frac 1 {\sqrt a} \int_0^\infty e^{-t^2} \rd t
| c = [[Integration by Substitution|substituting]] $t = \sqrt a x$
}}
{{eqn | r = \frac 1 2 \sqrt {\frac \pi a}
| c = [[Integral to Infinity of Ex... | Definite Integral to Infinity of Exponential of -a x^2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x^2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-a_x^2 | [
"Definite Integrals involving Exponential Function"
] | [] | [
"Integration by Substitution",
"Integral to Infinity of Exponential of -t^2"
] |
proofwiki-16208 | Definite Integral from 0 to 1 of Difference of Powers of x over Logarithm of x | :$\ds \int_0^1 \frac {x^m - x^n} {\ln x} \rd x = \map \ln {\frac {m + 1} {n + 1} }$ | Let:
:$x = e^{-u}$
We have, by Derivative of Exponential Function:
:$\dfrac {\d x} {\d u} = -e^{-u}$
By Exponential Tends to Zero and Infinity:
:as $x \to 0$, $u \to \infty$
By Exponential of Zero:
:as $x \to 1$, $u \to 0$.
So:
{{begin-eqn}}
{{eqn | l = \int_0^1 \frac {x^m - x^n} {\ln x} \rd x
| r = -\int_\infty^0... | :$\ds \int_0^1 \frac {x^m - x^n} {\ln x} \rd x = \map \ln {\frac {m + 1} {n + 1} }$ | Let:
:$x = e^{-u}$
We have, by [[Derivative of Exponential Function]]:
:$\dfrac {\d x} {\d u} = -e^{-u}$
By [[Exponential Tends to Zero and Infinity]]:
:as $x \to 0$, $u \to \infty$
By [[Exponential of Zero]]:
:as $x \to 1$, $u \to 0$.
So:
{{begin-eqn}}
{{eqn | l = \int_0^1 \frac {x^m - x^n} {\ln x} \rd x
... | Definite Integral from 0 to 1 of Difference of Powers of x over Logarithm of x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Difference_of_Powers_of_x_over_Logarithm_of_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Difference_of_Powers_of_x_over_Logarithm_of_x | [
"Definite Integrals involving Logarithm Function"
] | [] | [
"Derivative of Exponential Function",
"Exponential Tends to Zero and Infinity",
"Exponential of Zero",
"Integration by Substitution",
"Reversal of Limits of Definite Integral",
"Exponential of Sum",
"Definite Integral to Infinity of Exponential of -a x minus Exponential of -b x over x"
] |
proofwiki-16209 | Union is Commutative/Family of Sets | Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets.
Let $\ds I = \bigcup_{i \mathop \in I} S_i$ denote the union of $\family {S_i}_{i \mathop \in I}$.
Let $J \subseteq I$ be a subset of $I$.
Then:
:$\ds \bigcup_{i \mathop \in I} S_i = \bigcup_{j \mathop \in J} S_j \cup \bigcup_{k \mathop \in \relcomp I... | We have that both $\ds \bigcup_{j \mathop \in J} S_j$ and $\ds \bigcup_{k \mathop \in \relcomp I J} S_k$ are sets.
Hence by Union is Commutative we have:
:$\bigcup_{j \mathop \in J} S_j \cup \bigcup_{k \mathop \in \relcomp I J} S_k = \bigcup_{k \mathop \in \relcomp I J} S_k \cup \bigcup_{j \mathop \in J} S_j$
It remain... | Let $\family {S_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family of Sets|indexed family of sets]].
Let $\ds I = \bigcup_{i \mathop \in I} S_i$ denote the [[Definition:Union of Family|union]] of $\family {S_i}_{i \mathop \in I}$.
Let $J \subseteq I$ be a [[Definition:Subset|subset]] of $I$.
Then:
:$\ds \bigc... | We have that both $\ds \bigcup_{j \mathop \in J} S_j$ and $\ds \bigcup_{k \mathop \in \relcomp I J} S_k$ are [[Definition:Set|sets]].
Hence by [[Union is Commutative]] we have:
:$\bigcup_{j \mathop \in J} S_j \cup \bigcup_{k \mathop \in \relcomp I J} S_k = \bigcup_{k \mathop \in \relcomp I J} S_k \cup \bigcup_{j \math... | Union is Commutative/Family of Sets | https://proofwiki.org/wiki/Union_is_Commutative/Family_of_Sets | https://proofwiki.org/wiki/Union_is_Commutative/Family_of_Sets | [
"Union is Commutative"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Set Union/Family of Sets",
"Definition:Subset",
"Definition:Relative Complement"
] | [
"Definition:Set",
"Union is Commutative",
"Definition:Set Equality"
] |
proofwiki-16210 | Intersection is Commutative/Family of Sets | Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets.
Let $\ds I = \bigcap_{i \mathop \in I} S_i$ denote the intersection of $\family {S_i}_{i \mathop \in I}$.
Let $J \subseteq I$ be a subset of $I$.
Then:
:$\ds \bigcap_{i \mathop \in I} S_i = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \rel... | We have that both $\ds \bigcap_{j \mathop \in J} S_j$ and $\ds \bigcap_{k \mathop \in \relcomp I J} S_k$ are sets.
Hence by Intersection is Commutative we have:
:$\ds \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k = \bigcap_{k \mathop \in \relcomp I J} S_k \cap \bigcap_{j \mathop \in J} S_j... | Let $\family {S_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family of Sets|indexed family of sets]].
Let $\ds I = \bigcap_{i \mathop \in I} S_i$ denote the [[Definition:Intersection of Family|intersection]] of $\family {S_i}_{i \mathop \in I}$.
Let $J \subseteq I$ be a [[Definition:Subset|subset]] of $I$.
Then... | We have that both $\ds \bigcap_{j \mathop \in J} S_j$ and $\ds \bigcap_{k \mathop \in \relcomp I J} S_k$ are [[Definition:Set|sets]].
Hence by [[Intersection is Commutative]] we have:
:$\ds \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k = \bigcap_{k \mathop \in \relcomp I J} S_k \cap \bigc... | Intersection is Commutative/Family of Sets/Proof 1 | https://proofwiki.org/wiki/Intersection_is_Commutative/Family_of_Sets | https://proofwiki.org/wiki/Intersection_is_Commutative/Family_of_Sets/Proof_1 | [
"Intersection is Commutative"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Set Intersection/Family of Sets",
"Definition:Subset",
"Definition:Relative Complement"
] | [
"Definition:Set",
"Intersection is Commutative",
"Definition:Set Equality"
] |
proofwiki-16211 | Intersection is Commutative/Family of Sets | Let $\family {S_i}_{i \mathop \in I}$ be an indexed family of sets.
Let $\ds I = \bigcap_{i \mathop \in I} S_i$ denote the intersection of $\family {S_i}_{i \mathop \in I}$.
Let $J \subseteq I$ be a subset of $I$.
Then:
:$\ds \bigcap_{i \mathop \in I} S_i = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \rel... | {{begin-eqn}}
{{eqn | l = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k
| r = \map \complement {\map \complement {\bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k} }
| c = Complement of Complement
}}
{{eqn | r = \map \complement {\map \complement {\bi... | Let $\family {S_i}_{i \mathop \in I}$ be an [[Definition:Indexed Family of Sets|indexed family of sets]].
Let $\ds I = \bigcap_{i \mathop \in I} S_i$ denote the [[Definition:Intersection of Family|intersection]] of $\family {S_i}_{i \mathop \in I}$.
Let $J \subseteq I$ be a [[Definition:Subset|subset]] of $I$.
Then... | {{begin-eqn}}
{{eqn | l = \bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k
| r = \map \complement {\map \complement {\bigcap_{j \mathop \in J} S_j \cap \bigcap_{k \mathop \in \relcomp I J} S_k} }
| c = [[Complement of Complement]]
}}
{{eqn | r = \map \complement {\map \complement ... | Intersection is Commutative/Family of Sets/Proof 2 | https://proofwiki.org/wiki/Intersection_is_Commutative/Family_of_Sets | https://proofwiki.org/wiki/Intersection_is_Commutative/Family_of_Sets/Proof_2 | [
"Intersection is Commutative"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Set Intersection/Family of Sets",
"Definition:Subset",
"Definition:Relative Complement"
] | [
"Complement of Complement",
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Intersection",
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of Intersection",
"Union is Commutative/Family of Sets",
"De Morgan's Laws (Set Theory)/Set Complement/Family of Sets/Complement of I... |
proofwiki-16212 | Definite Integral to Infinity of Power of x by Logarithm of x over One plus x | :$\ds \int_0^\infty \frac {x^{p - 1} \ln x} {1 + x} \rd x = -\pi^2 \csc p \pi \cot p \pi$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {x^{p - 1} \ln x} {1 + x} \rd x
| r = \int_0^\infty \frac 1 {1 + x} \map {\frac \partial {\partial p} } {x^{p - 1} } \rd x
| c = Derivative of Power of Constant
}}
{{eqn | r = \frac \d {\d p} \int_0^\infty \frac {x^{p - 1} } {1 + x} \rd x
| c = Definite Integral of Parti... | :$\ds \int_0^\infty \frac {x^{p - 1} \ln x} {1 + x} \rd x = -\pi^2 \csc p \pi \cot p \pi$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {x^{p - 1} \ln x} {1 + x} \rd x
| r = \int_0^\infty \frac 1 {1 + x} \map {\frac \partial {\partial p} } {x^{p - 1} } \rd x
| c = [[Derivative of Power of Constant]]
}}
{{eqn | r = \frac \d {\d p} \int_0^\infty \frac {x^{p - 1} } {1 + x} \rd x
| c = [[Definite Integral of... | Definite Integral to Infinity of Power of x by Logarithm of x over One plus x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_by_Logarithm_of_x_over_One_plus_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_by_Logarithm_of_x_over_One_plus_x | [
"Definite Integrals involving Logarithm Function"
] | [] | [
"Derivative of General Exponential Function",
"Definite Integral of Partial Derivative",
"Definite Integral to Infinity of Power of x over 1 + x",
"Derivative of Composite Function",
"Derivative of Cosecant Function"
] |
proofwiki-16213 | Definite Integral from 0 to Half Pi of Logarithm of Sine x | :$\ds \int_0^{\pi/2} \map \ln {\sin x} \rd x = -\frac \pi 2 \ln 2$ | By Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\sin x}$: Lemma, we have:
:$\ds \int_0^\pi \map \ln {\sin x} \rd x = 2 \int_0^{\pi/2} \map \ln {\sin x} \rd x$
We also have:
{{begin-eqn}}
{{eqn | l = \int_0^{\pi/2} \map \ln {\sin x} \rd x
| r = \int_0^{\pi/2} \map \ln {\map \sin {\frac \pi 2 - x} } \rd ... | :$\ds \int_0^{\pi/2} \map \ln {\sin x} \rd x = -\frac \pi 2 \ln 2$ | By [[Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma|Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\sin x}$: Lemma]], we have:
:$\ds \int_0^\pi \map \ln {\sin x} \rd x = 2 \int_0^{\pi/2} \map \ln {\sin x} \rd x$
We also have:
{{begin-eqn}}
{{eqn | l = \int_0^{\pi/2} \map \ln {\sin x} ... | Definite Integral from 0 to Half Pi of Logarithm of Sine x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Logarithm_of_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Logarithm_of_Sine_x/Proof_1 | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Sine Function",
"Definite Integral from 0 to Half Pi of Logarithm of Sine x"
] | [] | [
"Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma",
"Integral between Limits is Independent of Direction",
"Sine of Complement equals Cosine",
"Sum of Logarithms",
"Double Angle Formulas/Sine",
"Primitive of Constant",
"Sum of Logarithms",
"Integration by Substitution",
"Logarithm of... |
proofwiki-16214 | Definite Integral from 0 to Half Pi of Logarithm of Sine x | :$\ds \int_0^{\pi/2} \map \ln {\sin x} \rd x = -\frac \pi 2 \ln 2$ | By {{Lemma|Definite Integral from 0 to Half Pi of Logarithm of Sine x|disp = Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\sin x}$}}, we have:
:$\ds \int_0^\pi \map \ln {\sin x} \rd x = 2 \int_0^{\pi/2} \map \ln {\sin x} \rd x$
By Product of Sines of Fractions of Pi, we have:
:$\ds \prod_{k \mathop = 1}... | :$\ds \int_0^{\pi/2} \map \ln {\sin x} \rd x = -\frac \pi 2 \ln 2$ | By {{Lemma|Definite Integral from 0 to Half Pi of Logarithm of Sine x|disp = Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\sin x}$}}, we have:
:$\ds \int_0^\pi \map \ln {\sin x} \rd x = 2 \int_0^{\pi/2} \map \ln {\sin x} \rd x$
By [[Product of Sines of Fractions of Pi]], we have:
:$\ds \prod_{k \math... | Definite Integral from 0 to Half Pi of Logarithm of Sine x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Logarithm_of_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Logarithm_of_Sine_x/Proof_2 | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Sine Function",
"Definite Integral from 0 to Half Pi of Logarithm of Sine x"
] | [] | [
"Product of Sines of Fractions of Pi",
"Sum of Logarithms/Natural Logarithm",
"Difference of Logarithms",
"Limit to Infinity of Power of x by Exponential of -a x",
"Combination Theorem for Limits of Functions/Real/Combined Sum Rule"
] |
proofwiki-16215 | Definite Integral from 0 to Half Pi of Logarithm of Cosine x | :$\ds \int_0^{\pi/2} \map \ln {\cos x} \rd x = -\frac \pi 2 \ln 2$ | By Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\sin x}$: Proof 1 we have:
:$\ds \int_0^{\pi/2} \map \ln {\sin x} \rd x = \int_0^{\pi/2} \map \ln {\cos x} \rd x$
and:
:$\ds \int_0^{\pi/2} \map \ln {\sin x} \rd x = -\frac \pi 2 \ln 2$
The result follows.
{{qed}} | :$\ds \int_0^{\pi/2} \map \ln {\cos x} \rd x = -\frac \pi 2 \ln 2$ | By [[Definite Integral from 0 to Half Pi of Logarithm of Sine x/Proof 1|Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\sin x}$: Proof 1]] we have:
:$\ds \int_0^{\pi/2} \map \ln {\sin x} \rd x = \int_0^{\pi/2} \map \ln {\cos x} \rd x$
and:
:$\ds \int_0^{\pi/2} \map \ln {\sin x} \rd x = -\frac \pi 2 \ln ... | Definite Integral from 0 to Half Pi of Logarithm of Cosine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Logarithm_of_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Logarithm_of_Cosine_x | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Cosine Function"
] | [] | [
"Definite Integral from 0 to Half Pi of Logarithm of Sine x/Proof 1"
] |
proofwiki-16216 | Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary/P-adic Norm | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\sequence {x_n}$ be a Cauchy sequence such that $\sequence {x_n}$ does not converge to $0$.
Then:
:$\exists N \in \N: \forall n, m \ge N: \norm {x_n}_p = \norm {x_m}_p$ | From:
:P-adic Numbers form Non-Archimedean Valued Field
:Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary
it follows that:
:$\exists N \in \N: \forall n, m \ge N: \norm {x_n}_p = \norm {x_m}_p$
{{qed}} | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\sequence {x_n}$ be a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] such that $\sequence {x_n}$ does not [[Definition:Co... | From:
:[[P-adic Numbers form Non-Archimedean Valued Field]]
:[[Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary]]
it follows that:
:$\exists N \in \N: \forall n, m \ge N: \norm {x_n}_p = \norm {x_m}_p$
{{qed}} | Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary/P-adic Norm | https://proofwiki.org/wiki/Non-Null_Cauchy_Sequence_in_Non-Archimedean_Norm_is_Eventually_Stationary/P-adic_Norm | https://proofwiki.org/wiki/Non-Null_Cauchy_Sequence_in_Non-Archimedean_Norm_is_Eventually_Stationary/P-adic_Norm | [
"Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Convergent Sequence/Normed Division Ring"
] | [
"P-adic Norm forms Non-Archimedean Valued Field/P-adic Numbers",
"Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary"
] |
proofwiki-16217 | Definite Integral to Infinity of Logarithm of Exponential of x plus One over Exponential of x minus One | :$\ds \int_0^\infty \map \ln {\frac {e^x + 1} {e^x - 1} } \rd x = \frac {\pi^2} 4$ | We can write:
{{begin-eqn}}
{{eqn | l = \int_0^\infty \map \ln {\frac {e^x + 1} {e^x - 1} } \rd x
| r = \int_0^\infty \map \ln {\frac {e^{x/2} \paren {e^{x/2} + e^{-x/2} } } {e^{x/2} \paren {e^{x/2} - e^{-x/2} } } } \rd x
}}
{{eqn | r = \int_0^\infty \map \ln {\coth \frac x 2} \rd x
| c = {{Defof|Hyperbolic Cotangen... | :$\ds \int_0^\infty \map \ln {\frac {e^x + 1} {e^x - 1} } \rd x = \frac {\pi^2} 4$ | We can write:
{{begin-eqn}}
{{eqn | l = \int_0^\infty \map \ln {\frac {e^x + 1} {e^x - 1} } \rd x
| r = \int_0^\infty \map \ln {\frac {e^{x/2} \paren {e^{x/2} + e^{-x/2} } } {e^{x/2} \paren {e^{x/2} - e^{-x/2} } } } \rd x
}}
{{eqn | r = \int_0^\infty \map \ln {\coth \frac x 2} \rd x
| c = {{Defof|Hyperbolic Cotange... | Definite Integral to Infinity of Logarithm of Exponential of x plus One over Exponential of x minus One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Logarithm_of_Exponential_of_x_plus_One_over_Exponential_of_x_minus_One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Logarithm_of_Exponential_of_x_plus_One_over_Exponential_of_x_minus_One | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Exponential Function"
] | [] | [
"Derivative of Hyperbolic Cotangent Function",
"Difference of Squares of Hyperbolic Cotangent and Cosecant",
"Limit to Infinity of Hyperbolic Cotangent Function",
"Integration by Substitution",
"Reversal of Limits of Definite Integral",
"Integration by Substitution",
"Reversal of Limits of Definite Inte... |
proofwiki-16218 | Definite Integral from 0 to Quarter Pi of Logarithm of One plus Tan x | :$\ds \int_0^{\pi/4} \map \ln {1 + \tan x} \rd x = \frac \pi 8 \ln 2$ | {{begin-eqn}}
{{eqn | l = \int_0^{\pi/4} \map \ln {1 + \tan x} \rd x
| r = \int_0^{\pi/4} \map \ln {1 + \map \tan {\frac \pi 4 - x} } \rd x
| c = Integral between Limits is Independent of Direction
}}
{{eqn | r = \int_0^{\pi/4} \map \ln {1 + \frac {\tan \frac \pi 4 - \tan x} {1 + \tan \frac \pi 4 \tan x} } \rd x
| c... | :$\ds \int_0^{\pi/4} \map \ln {1 + \tan x} \rd x = \frac \pi 8 \ln 2$ | {{begin-eqn}}
{{eqn | l = \int_0^{\pi/4} \map \ln {1 + \tan x} \rd x
| r = \int_0^{\pi/4} \map \ln {1 + \map \tan {\frac \pi 4 - x} } \rd x
| c = [[Integral between Limits is Independent of Direction]]
}}
{{eqn | r = \int_0^{\pi/4} \map \ln {1 + \frac {\tan \frac \pi 4 - \tan x} {1 + \tan \frac \pi 4 \tan x} } \rd x
... | Definite Integral from 0 to Quarter Pi of Logarithm of One plus Tan x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Quarter_Pi_of_Logarithm_of_One_plus_Tan_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Quarter_Pi_of_Logarithm_of_One_plus_Tan_x | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Tangent Function"
] | [] | [
"Integral between Limits is Independent of Direction",
"Tangent of Difference",
"Tangent of 45 Degrees",
"Difference of Logarithms",
"Primitive of Constant"
] |
proofwiki-16219 | Primitive of Sine x by Logarithm of Sine x | :$\ds \int \sin x \map \ln {\sin x} \rd x = \cos x \paren {1 - \map \ln {\sin x} } + \ln \size {\tan \frac x 2} + C$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\map \ln {\sin x} }
| r = \map {\frac \d {\map \d {\sin x} } } {\map \ln {\sin x} } \map {\frac \d {\d x} } {\sin x}
| c = Chain Rule for Derivatives
}}
{{eqn | r = \frac {\cos x} {\sin x}
| c = Derivative of Logarithm Function, Derivative of Sine Functio... | :$\ds \int \sin x \map \ln {\sin x} \rd x = \cos x \paren {1 - \map \ln {\sin x} } + \ln \size {\tan \frac x 2} + C$ | We have:
{{begin-eqn}}
{{eqn | l = \map {\frac \d {\d x} } {\map \ln {\sin x} }
| r = \map {\frac \d {\map \d {\sin x} } } {\map \ln {\sin x} } \map {\frac \d {\d x} } {\sin x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | r = \frac {\cos x} {\sin x}
| c = [[Derivative of Logarithm Function]], [[Derivative of S... | Primitive of Sine x by Logarithm of Sine x | https://proofwiki.org/wiki/Primitive_of_Sine_x_by_Logarithm_of_Sine_x | https://proofwiki.org/wiki/Primitive_of_Sine_x_by_Logarithm_of_Sine_x | [
"Primitives involving Sine Function",
"Primitives involving Logarithm Function"
] | [] | [
"Derivative of Composite Function",
"Derivative of Logarithm Function",
"Derivative of Sine Function",
"Primitive of Sine Function",
"Integration by Parts",
"Sum of Squares of Sine and Cosine",
"Primitive of Sine Function",
"Primitive of Cosecant Function/Tangent Form",
"Category:Primitives involvin... |
proofwiki-16220 | Euler-Mascheroni Constant as Difference of Integrals involving Cosine | :$\ds \int_0^1 \frac {1 - \cos x} x \rd x - \int_1^\infty \frac {\cos x} x \rd x = \gamma$ | From Characterization of Cosine Integral Function, we have:
:$\ds \int_t^\infty \frac {\cos x} x \rd x = -\gamma - \ln t + \int_0^t \frac {1 - \cos x} x \rd x$
for all real $t > 0$.
Rearranging:
:$\ds \int_0^t \frac {1 - \cos x} x \rd x - \int_t^\infty \frac {\cos x} x \rd x = \gamma + \ln t$
Setting $t = 1$ gives th... | :$\ds \int_0^1 \frac {1 - \cos x} x \rd x - \int_1^\infty \frac {\cos x} x \rd x = \gamma$ | From [[Characterization of Cosine Integral Function]], we have:
:$\ds \int_t^\infty \frac {\cos x} x \rd x = -\gamma - \ln t + \int_0^t \frac {1 - \cos x} x \rd x$
for all [[Definition:Real Number|real]] $t > 0$.
Rearranging:
:$\ds \int_0^t \frac {1 - \cos x} x \rd x - \int_t^\infty \frac {\cos x} x \rd x = \gamm... | Euler-Mascheroni Constant as Difference of Integrals involving Cosine | https://proofwiki.org/wiki/Euler-Mascheroni_Constant_as_Difference_of_Integrals_involving_Cosine | https://proofwiki.org/wiki/Euler-Mascheroni_Constant_as_Difference_of_Integrals_involving_Cosine | [
"Definite Integrals involving Cosine Function",
"Euler-Mascheroni Constant"
] | [] | [
"Characterization of Cosine Integral Function",
"Definition:Real Number",
"Natural Logarithm of 1 is 0"
] |
proofwiki-16221 | Definite Integral from 0 to 1 of Arcsine of x over x | :$\ds \int_0^1 \frac {\arcsin x} x = \frac \pi 2 \ln 2$ | Let:
:$x = \sin \theta$
By Derivative of Sine Function, we have:
:$\dfrac {\d x} {\d \theta} = \cos \theta$
We have, by Arcsine of Zero is Zero:
:as $x \to 0$, $\theta \to \arcsin 0 = 0$.
By Arcsine of One is Half Pi, we have:
:as $x \to 1$, $\theta \to \arcsin 1 = \dfrac \pi 2$.
We have:
{{begin-eqn}}
{{eqn | l = ... | :$\ds \int_0^1 \frac {\arcsin x} x = \frac \pi 2 \ln 2$ | Let:
:$x = \sin \theta$
By [[Derivative of Sine Function]], we have:
:$\dfrac {\d x} {\d \theta} = \cos \theta$
We have, by [[Arcsine of Zero is Zero]]:
:as $x \to 0$, $\theta \to \arcsin 0 = 0$.
By [[Arcsine of One is Half Pi]], we have:
:as $x \to 1$, $\theta \to \arcsin 1 = \dfrac \pi 2$.
We have:
{{beg... | Definite Integral from 0 to 1 of Arcsine of x over x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Arcsine_of_x_over_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Arcsine_of_x_over_x | [
"Definite Integrals involving Arcsine Function"
] | [] | [
"Derivative of Sine Function",
"Arcsine of Zero is Zero",
"Arcsine of One is Half Pi",
"Integration by Substitution",
"Primitive of Cotangent Function",
"Integration by Parts",
"Definite Integral from 0 to Half Pi of Logarithm of Sine x",
"Sine of Right Angle",
"Natural Logarithm of 1 is 0",
"Sum ... |
proofwiki-16222 | Definite Integral from 0 to 1 of Arctangent of x over x | :$\ds \int_0^1 \frac {\arctan x} x \rd x = G$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\arctan x} x \rd x
| r = \int_0^1 \frac 1 x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} } \rd x
| c = Power Series Expansion for Real Arctangent Function
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {2 n + 1} ... | :$\ds \int_0^1 \frac {\arctan x} x \rd x = G$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {\arctan x} x \rd x
| r = \int_0^1 \frac 1 x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} } \rd x
| c = [[Power Series Expansion for Real Arctangent Function]]
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {2 n +... | Definite Integral from 0 to 1 of Arctangent of x over x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Arctangent_of_x_over_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Arctangent_of_x_over_x | [
"Definite Integrals involving Arctangent Function"
] | [] | [
"Power Series Expansion for Real Arctangent Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Primitive of Power",
"Definition:Definite Integral"
] |
proofwiki-16223 | Definite Integral from 0 to Half Pi of x over Sine x | :$\ds \int_0^{\pi/2} \frac x {\sin x} \rd x = 2 G$ | From Definite Integral from $0$ to $1$ of $\dfrac {\arctan x} x$, we have:
:$\ds \int_0^1 \frac {\arctan x} x \rd x = G$
Let:
:$x = \tan \theta$
By Derivative of Tangent Function, we have:
:$\ds \frac {\d x} {\d \theta} = \sec^2 \theta$
We have, by Arctangent of Zero is Zero:
:as $x \to 0$, $\theta \to 0$.
We also... | :$\ds \int_0^{\pi/2} \frac x {\sin x} \rd x = 2 G$ | From [[Definite Integral from 0 to 1 of Arctangent of x over x|Definite Integral from $0$ to $1$ of $\dfrac {\arctan x} x$]], we have:
:$\ds \int_0^1 \frac {\arctan x} x \rd x = G$
Let:
:$x = \tan \theta$
By [[Derivative of Tangent Function]], we have:
:$\ds \frac {\d x} {\d \theta} = \sec^2 \theta$
We have, ... | Definite Integral from 0 to Half Pi of x over Sine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_x_over_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_x_over_Sine_x | [
"Definite Integrals involving Sine Function"
] | [] | [
"Definite Integral from 0 to 1 of Arctangent of x over x",
"Derivative of Tangent Function",
"Arctangent of Zero is Zero",
"Arctangent of One",
"Integration by Substitution",
"Double Angle Formulas/Sine",
"Integration by Substitution"
] |
proofwiki-16224 | Definite Integral to Infinity of Arctangent of p x minus Arctangent of q x over x | :$\ds \int_0^\infty \frac {\arctan p x - \arctan q x} x \rd x = \frac \pi 2 \ln \frac p q$ | Note that the integrand is of the form:
:$\ds \int_0^\infty \frac {\map f {p x} - \map f {q x} } x \rd x$
where:
:$\map f x = \arctan x$
We have, by Derivative of Arctangent Function:
:$\map {f'} x = \dfrac 1 {1 + x^2}$
which is continuous on $\R$.
By Limit to Infinity of Arctangent Function:
:$\ds \lim_{x \mathop \... | :$\ds \int_0^\infty \frac {\arctan p x - \arctan q x} x \rd x = \frac \pi 2 \ln \frac p q$ | Note that the [[Definition:Integrand|integrand]] is of the form:
:$\ds \int_0^\infty \frac {\map f {p x} - \map f {q x} } x \rd x$
where:
:$\map f x = \arctan x$
We have, by [[Derivative of Arctangent Function]]:
:$\map {f'} x = \dfrac 1 {1 + x^2}$
which is [[Definition:Continuous Mapping|continuous]] on $\R$.
... | Definite Integral to Infinity of Arctangent of p x minus Arctangent of q x over x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Arctangent_of_p_x_minus_Arctangent_of_q_x_over_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Arctangent_of_p_x_minus_Arctangent_of_q_x_over_x | [
"Definite Integrals involving Arctangent Function"
] | [] | [
"Definition:Integration/Integrand",
"Derivative of Arctangent Function",
"Definition:Continuous Mapping",
"Limit to Infinity of Arctangent Function",
"Definition:Continuously Differentiable",
"Frullani's Integral",
"Arctangent of Zero is Zero"
] |
proofwiki-16225 | Definite Integral from 0 to Half Pi of Sine x by Logarithm of Sine x | :$\ds \int_0^{\pi/2} \sin x \map \ln {\sin x} \rd x = \ln 2 - 1$ | {{begin-eqn}}
{{eqn | l = \int_0^{\pi/2} \sin x \map \ln {\sin x} \rd x
| r = \intlimits {\cos x \paren {1 - \map \ln {\sin x} } + \map \ln {\tan \frac x 2} } 0 {\pi/2}
| c = Primitive of $\sin x \map \ln {\sin x}$
}}
{{eqn | r = \cos \frac \pi 2 \paren {1 - \map \ln {\sin \frac \pi 2} } + \map \ln {\tan \frac \pi 4}... | :$\ds \int_0^{\pi/2} \sin x \map \ln {\sin x} \rd x = \ln 2 - 1$ | {{begin-eqn}}
{{eqn | l = \int_0^{\pi/2} \sin x \map \ln {\sin x} \rd x
| r = \intlimits {\cos x \paren {1 - \map \ln {\sin x} } + \map \ln {\tan \frac x 2} } 0 {\pi/2}
| c = [[Primitive of Sine x by Logarithm of Sine x|Primitive of $\sin x \map \ln {\sin x}$]]
}}
{{eqn | r = \cos \frac \pi 2 \paren {1 - \map \ln {\s... | Definite Integral from 0 to Half Pi of Sine x by Logarithm of Sine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Sine_x_by_Logarithm_of_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Sine_x_by_Logarithm_of_Sine_x | [
"Definite Integrals involving Sine Function",
"Definite Integrals involving Logarithm Function"
] | [] | [
"Primitive of Sine x by Logarithm of Sine x",
"Cosine of Right Angle",
"Tangent of 45 Degrees",
"Natural Logarithm of 1 is 0",
"Cosine of Zero is One",
"Double Angle Formulas/Sine",
"Difference of Logarithms",
"Sum of Logarithms",
"Combination Theorem for Limits of Functions/Real/Sum Rule",
"Cosin... |
proofwiki-16226 | Definite Integral from 0 to Pi of x by Logarithm of Sine x | :$\ds \int_0^\pi x \map \ln {\sin x} \rd x = -\frac {\pi^2} 2 \ln 2$ | {{begin-eqn}}
{{eqn | l = \int_0^\pi x \map \ln {\sin x} \rd x
| r = \int_0^\pi \paren {\pi - x} \map \ln {\map \sin {\pi - x} } \rd x
| c = Integral between Limits is Independent of Direction
}}
{{eqn | r = \pi \int_0^\pi \map \ln {\sin x} - \int_0^\pi x \map \ln {\sin x} \rd x
| c = Sine of Supplementary Angle, Li... | :$\ds \int_0^\pi x \map \ln {\sin x} \rd x = -\frac {\pi^2} 2 \ln 2$ | {{begin-eqn}}
{{eqn | l = \int_0^\pi x \map \ln {\sin x} \rd x
| r = \int_0^\pi \paren {\pi - x} \map \ln {\map \sin {\pi - x} } \rd x
| c = [[Integral between Limits is Independent of Direction]]
}}
{{eqn | r = \pi \int_0^\pi \map \ln {\sin x} - \int_0^\pi x \map \ln {\sin x} \rd x
| c = [[Sine of Supplementary Ang... | Definite Integral from 0 to Pi of x by Logarithm of Sine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_x_by_Logarithm_of_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_x_by_Logarithm_of_Sine_x | [
"Definite Integrals involving Logarithm Function",
"Definite Integrals involving Sine Function"
] | [] | [
"Integral between Limits is Independent of Direction",
"Sine of Supplementary Angle",
"Linear Combination of Integrals/Definite",
"Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma",
"Definite Integral from 0 to Half Pi of Logarithm of Sine x"
] |
proofwiki-16227 | Definite Integral to Infinity of Exponential of -(a x^2 plus b x plus c) | :$\ds \int_0^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} } \map \erfc {\frac b {2 \sqrt a} }$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \map \exp {-\paren {a x^2 + bx + c} } \rd x
| r = \int_0^\infty \map \exp {-a \paren {x + \frac b {2 a} }^2 + \frac {b^2} {4 a} - c} \rd x
| c = Completing the Square
}}
{{eqn | r = \map \exp {\frac {b^2 - 4 a c} {4 a} } \int_0^\infty \map \exp {-a \paren {x + \frac b {2 a} }^2... | :$\ds \int_0^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} } \map \erfc {\frac b {2 \sqrt a} }$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \map \exp {-\paren {a x^2 + bx + c} } \rd x
| r = \int_0^\infty \map \exp {-a \paren {x + \frac b {2 a} }^2 + \frac {b^2} {4 a} - c} \rd x
| c = [[Completing the Square]]
}}
{{eqn | r = \map \exp {\frac {b^2 - 4 a c} {4 a} } \int_0^\infty \map \exp {-a \paren {x + \frac b {2 a}... | Definite Integral to Infinity of Exponential of -(a x^2 plus b x plus c) | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-(a_x^2_plus_b_x_plus_c) | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-(a_x^2_plus_b_x_plus_c) | [
"Definite Integrals involving Exponential Function"
] | [] | [
"Completing the Square",
"Exponential of Sum",
"Integration by Substitution"
] |
proofwiki-16228 | Set is Subset of Intersection of Supersets | Let $S$, $T_1$ and $T_2$ be sets.
Let $S$ be a subset of both $T_1$ and $T_2$.
Then:
:$S \subseteq T_1 \cap T_2$
That is:
:$\paren {S \subseteq T_1} \land \paren {S \subseteq T_2} \implies S \subseteq \paren {T_1 \cap T_2}$ | Let $S \subseteq T_1 \land S \subseteq T_2$.
Then:
{{begin-eqn}}
{{eqn | l = x \in S
| o = \leadsto
| r = x \in T_1 \land x \in T_2
| c = {{Defof|Subset}}
}}
{{eqn | o = \leadsto
| r = x \in T_1 \cap T_2
| c = {{Defof|Set Intersection}}
}}
{{eqn | o = \leadsto
| r = S \subseteq T_1 \... | Let $S$, $T_1$ and $T_2$ be [[Definition:Set|sets]].
Let $S$ be a [[Definition:Subset|subset]] of both $T_1$ and $T_2$.
Then:
:$S \subseteq T_1 \cap T_2$
That is:
:$\paren {S \subseteq T_1} \land \paren {S \subseteq T_2} \implies S \subseteq \paren {T_1 \cap T_2}$ | Let $S \subseteq T_1 \land S \subseteq T_2$.
Then:
{{begin-eqn}}
{{eqn | l = x \in S
| o = \leadsto
| r = x \in T_1 \land x \in T_2
| c = {{Defof|Subset}}
}}
{{eqn | o = \leadsto
| r = x \in T_1 \cap T_2
| c = {{Defof|Set Intersection}}
}}
{{eqn | o = \leadsto
| r = S \subseteq T_... | Set is Subset of Intersection of Supersets/Proof 1 | https://proofwiki.org/wiki/Set_is_Subset_of_Intersection_of_Supersets | https://proofwiki.org/wiki/Set_is_Subset_of_Intersection_of_Supersets/Proof_1 | [
"Set Intersection",
"Subsets",
"Set is Subset of Intersection of Supersets"
] | [
"Definition:Set",
"Definition:Subset"
] | [] |
proofwiki-16229 | Set is Subset of Intersection of Supersets | Let $S$, $T_1$ and $T_2$ be sets.
Let $S$ be a subset of both $T_1$ and $T_2$.
Then:
:$S \subseteq T_1 \cap T_2$
That is:
:$\paren {S \subseteq T_1} \land \paren {S \subseteq T_2} \implies S \subseteq \paren {T_1 \cap T_2}$ | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T_1
}}
{{eqn | lo= \land
| l = S
| o = \subseteq
| r = T_2
}}
{{eqn | ll= \leadsto
| l = S \cap S
| o = \subseteq
| r = S \cap T_2
| c = Set Intersection Preserves Subsets
}}
{{eqn | ll= \leadsto
| l = S
... | Let $S$, $T_1$ and $T_2$ be [[Definition:Set|sets]].
Let $S$ be a [[Definition:Subset|subset]] of both $T_1$ and $T_2$.
Then:
:$S \subseteq T_1 \cap T_2$
That is:
:$\paren {S \subseteq T_1} \land \paren {S \subseteq T_2} \implies S \subseteq \paren {T_1 \cap T_2}$ | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T_1
}}
{{eqn | lo= \land
| l = S
| o = \subseteq
| r = T_2
}}
{{eqn | ll= \leadsto
| l = S \cap S
| o = \subseteq
| r = S \cap T_2
| c = [[Set Intersection Preserves Subsets]]
}}
{{eqn | ll= \leadsto
| l = S
... | Set is Subset of Intersection of Supersets/Proof 2 | https://proofwiki.org/wiki/Set_is_Subset_of_Intersection_of_Supersets | https://proofwiki.org/wiki/Set_is_Subset_of_Intersection_of_Supersets/Proof_2 | [
"Set Intersection",
"Subsets",
"Set is Subset of Intersection of Supersets"
] | [
"Definition:Set",
"Definition:Subset"
] | [
"Set Intersection Preserves Subsets",
"Set Intersection is Idempotent"
] |
proofwiki-16230 | Definite Integral over Reals of Exponential of -(a x^2 plus b x plus c) | :$\ds \int_{-\infty}^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x = \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} }$ | {{begin-eqn}}
{{eqn | l = \int_{-\infty}^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x
| r = \int_{-\infty}^\infty \map \exp {-a \paren {x + \frac b {2 a} }^2 + \frac {b^2} {4 a} - c} \rd x
| c = Completing the Square
}}
{{eqn | r = \map \exp {\frac {b^2 - 4 a c} {4 a} } \int_{-\infty}^\infty \map \exp {-a \par... | :$\ds \int_{-\infty}^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x = \sqrt {\frac \pi a} \map \exp {\frac {b^2 - 4 a c} {4 a} }$ | {{begin-eqn}}
{{eqn | l = \int_{-\infty}^\infty \map \exp {-\paren {a x^2 + b x + c} } \rd x
| r = \int_{-\infty}^\infty \map \exp {-a \paren {x + \frac b {2 a} }^2 + \frac {b^2} {4 a} - c} \rd x
| c = [[Completing the Square]]
}}
{{eqn | r = \map \exp {\frac {b^2 - 4 a c} {4 a} } \int_{-\infty}^\infty \map \exp {-a ... | Definite Integral over Reals of Exponential of -(a x^2 plus b x plus c) | https://proofwiki.org/wiki/Definite_Integral_over_Reals_of_Exponential_of_-(a_x^2_plus_b_x_plus_c) | https://proofwiki.org/wiki/Definite_Integral_over_Reals_of_Exponential_of_-(a_x^2_plus_b_x_plus_c) | [
"Definite Integrals involving Exponential Function"
] | [] | [
"Completing the Square",
"Exponential of Sum",
"Integration by Substitution",
"Gaussian Integral"
] |
proofwiki-16231 | Definite Integral to Infinity of Power of x by Exponential of -a x^2 | :$\ds \int_0^\infty x^m e^{-a x^2} \rd x = \frac {\map \Gamma {\paren {m + 1}/2} } {2 a^{\paren {m + 1}/2} }$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty x^m e^{-a x^2} \rd x
| r = \int_0^\infty \frac 1 {2 \sqrt t} \paren {\sqrt t}^m e^{-a t} \rd t
| c = substituting $t = x^2$
}}
{{eqn | r = \frac 1 2 \int_0^\infty t^{\paren {\paren {m + 1}/2} - 1} e^{-a t} \rd t
}}
{{eqn | r = \frac {\map \Gamma {\paren {m + 1}/2} } {... | :$\ds \int_0^\infty x^m e^{-a x^2} \rd x = \frac {\map \Gamma {\paren {m + 1}/2} } {2 a^{\paren {m + 1}/2} }$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty x^m e^{-a x^2} \rd x
| r = \int_0^\infty \frac 1 {2 \sqrt t} \paren {\sqrt t}^m e^{-a t} \rd t
| c = [[Integration by Substitution|substituting]] $t = x^2$
}}
{{eqn | r = \frac 1 2 \int_0^\infty t^{\paren {\paren {m + 1}/2} - 1} e^{-a t} \rd t
}}
{{eqn | r = \frac {\m... | Definite Integral to Infinity of Power of x by Exponential of -a x^2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_by_Exponential_of_-a_x^2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_by_Exponential_of_-a_x^2 | [
"Definite Integrals involving Exponential Function"
] | [] | [
"Integration by Substitution",
"Laplace Transform of Real Power"
] |
proofwiki-16232 | Definite Integral to Infinity of Exponential of -(a x^2 plus b over x^2) | :$\ds \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-2 \sqrt {a b} }$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x
| r = \int_0^\infty \map \exp {-a \paren {x^2 + \frac b {a x^2} } } \rd x
}}
{{eqn | r = \int_0^\infty \map \exp {-a \paren {\paren {x - \frac 1 x \sqrt {\frac b a} }^2 + 2 \sqrt {\frac b a} } } \rd x
| c = Completing the Squa... | :$\ds \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x = \frac 1 2 \sqrt {\frac \pi a} \map \exp {-2 \sqrt {a b} }$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \map \exp {-\paren {a x^2 + \frac b {x^2} } } \rd x
| r = \int_0^\infty \map \exp {-a \paren {x^2 + \frac b {a x^2} } } \rd x
}}
{{eqn | r = \int_0^\infty \map \exp {-a \paren {\paren {x - \frac 1 x \sqrt {\frac b a} }^2 + 2 \sqrt {\frac b a} } } \rd x
| c = [[Completing the Sq... | Definite Integral to Infinity of Exponential of -(a x^2 plus b over x^2) | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-(a_x^2_plus_b_over_x^2) | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-(a_x^2_plus_b_over_x^2) | [
"Definite Integrals involving Exponential Function"
] | [] | [
"Completing the Square",
"Exponential of Sum",
"Definite Integral of Even Function",
"Glasser's Master Theorem",
"Definite Integral of Even Function",
"Definite Integral to Infinity of Exponential of -a x^2"
] |
proofwiki-16233 | Definite Integral to Infinity of x over Exponential of x minus One | :$\ds \int_0^\infty \frac x {e^x - 1} \rd x = \frac {\pi^2} 6$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac x {e^x - 1} \rd x
| r = \int_0^\infty \frac {x^{2 - 1} } {e^x - 1} \rd x
}}
{{eqn | r = \map \zeta 2 \map \Gamma 2
| c = Integral Representation of Riemann Zeta Function in terms of Gamma Function
}}
{{eqn | r = \frac {\pi^2} 6 \times 1!
| c = Basel Problem, Gamma Functi... | :$\ds \int_0^\infty \frac x {e^x - 1} \rd x = \frac {\pi^2} 6$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac x {e^x - 1} \rd x
| r = \int_0^\infty \frac {x^{2 - 1} } {e^x - 1} \rd x
}}
{{eqn | r = \map \zeta 2 \map \Gamma 2
| c = [[Integral Representation of Riemann Zeta Function in terms of Gamma Function]]
}}
{{eqn | r = \frac {\pi^2} 6 \times 1!
| c = [[Basel Problem]], [[Ga... | Definite Integral to Infinity of x over Exponential of x minus One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_x_over_Exponential_of_x_minus_One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_x_over_Exponential_of_x_minus_One | [
"Definite Integrals involving Exponential Function"
] | [] | [
"Integral Representation of Riemann Zeta Function in terms of Gamma Function",
"Basel Problem",
"Gamma Function Extends Factorial"
] |
proofwiki-16234 | Integral Representation of Dirichlet Eta Function in terms of Gamma Function | :$\ds \map \eta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x
| r = \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 - \paren {-e^{-x} } } \rd x
}}
{{eqn | r = \int_0^\infty x^{s - 1} e^{-x} \paren {\sum_{n \mathop = 0}^\infty \paren {-e^{-x} }^n} \rd x
| c = Sum of Infinite Geometric Sequence
}}
{{eqn | r = \... | :$\ds \map \eta s = \frac 1 {\map \Gamma s} \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {x^{s - 1} } {e^x + 1} \rd x
| r = \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 - \paren {-e^{-x} } } \rd x
}}
{{eqn | r = \int_0^\infty x^{s - 1} e^{-x} \paren {\sum_{n \mathop = 0}^\infty \paren {-e^{-x} }^n} \rd x
| c = [[Sum of Infinite Geometric Sequence]]
}}
{{eqn | r... | Integral Representation of Dirichlet Eta Function in terms of Gamma Function | https://proofwiki.org/wiki/Integral_Representation_of_Dirichlet_Eta_Function_in_terms_of_Gamma_Function | https://proofwiki.org/wiki/Integral_Representation_of_Dirichlet_Eta_Function_in_terms_of_Gamma_Function | [
"Definite Integrals involving Exponential Function",
"Dirichlet Eta Function",
"Gamma Function"
] | [] | [
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Laplace Transform of Complex Power"
] |
proofwiki-16235 | Definite Integral to Infinity of x over Exponential of x plus One | :$\ds \int_0^\infty \frac x {e^x + 1} \rd x = \frac {\pi^2} {12}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac x {e^x + 1} \rd x
| r = \int_0^\infty \frac {x^{2 - 1} } {e^x + 1} \rd x
}}
{{eqn | r = \map \eta 2 \map \Gamma 2
| c = Integral Representation of Dirichlet Eta Function in terms of Gamma Function
}}
{{eqn | r = \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - ... | :$\ds \int_0^\infty \frac x {e^x + 1} \rd x = \frac {\pi^2} {12}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac x {e^x + 1} \rd x
| r = \int_0^\infty \frac {x^{2 - 1} } {e^x + 1} \rd x
}}
{{eqn | r = \map \eta 2 \map \Gamma 2
| c = [[Integral Representation of Dirichlet Eta Function in terms of Gamma Function]]
}}
{{eqn | r = \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{... | Definite Integral to Infinity of x over Exponential of x plus One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_x_over_Exponential_of_x_plus_One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_x_over_Exponential_of_x_plus_One | [
"Definite Integrals involving Exponential Function"
] | [] | [
"Integral Representation of Dirichlet Eta Function in terms of Gamma Function",
"Gamma Function Extends Factorial",
"Sum of Reciprocals of Squares Alternating in Sign"
] |
proofwiki-16236 | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x | :$\ds \int_0^1 \ln x \map \ln {1 - x} \rd x = 2 - \frac {\pi^2} 6$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \ln x \map \ln {1 - x} \rd x
| r = -\int_0^1 \ln x \paren {\sum_{n \mathop = 1}^\infty \frac {x^n} n} \rd x
| c = Power Series Expansion for $\map \ln {1 - x}$
}}
{{eqn | r = -\sum_{n \mathop = 1}^\infty \frac 1 n \paren {\int_0^1 x^n \ln x \rd x}
| c = Fubini's Theo... | :$\ds \int_0^1 \ln x \map \ln {1 - x} \rd x = 2 - \frac {\pi^2} 6$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \ln x \map \ln {1 - x} \rd x
| r = -\int_0^1 \ln x \paren {\sum_{n \mathop = 1}^\infty \frac {x^n} n} \rd x
| c = [[Power Series Expansion for Logarithm of 1 - x|Power Series Expansion for $\map \ln {1 - x}$]]
}}
{{eqn | r = -\sum_{n \mathop = 1}^\infty \frac 1 n \paren {\... | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_minus_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_minus_x | [
"Definite Integrals involving Logarithm Function"
] | [] | [
"Power Series Expansion for Logarithm of 1 - x",
"Fubini's Theorem",
"Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x",
"Gamma Function Extends Factorial",
"Telescoping Series/Example 1",
"Basel Problem"
] |
proofwiki-16237 | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x | :$\ds \int_0^1 \ln x \map \ln {1 - x} \rd x = 2 - \frac {\pi^2} 6$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \dfrac {\ln x \map \ln {1 - x} } {\paren {1 - x} } \rd x
| r = \int_1^0 \dfrac {\map \ln {1 - u} \ln u } u \paren {-\rd u}
| c = $x \to \paren {1 - u}$ and $\rd x \to -\rd u$
}}
{{eqn | r = \int_0^1 \dfrac {\map \ln {1 - u} \ln u } u \rd u
| c = reversing limits of inte... | :$\ds \int_0^1 \ln x \map \ln {1 - x} \rd x = 2 - \frac {\pi^2} 6$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \dfrac {\ln x \map \ln {1 - x} } {\paren {1 - x} } \rd x
| r = \int_1^0 \dfrac {\map \ln {1 - u} \ln u } u \paren {-\rd u}
| c = $x \to \paren {1 - u}$ and $\rd x \to -\rd u$
}}
{{eqn | r = \int_0^1 \dfrac {\map \ln {1 - u} \ln u } u \rd u
| c = [[Integration by Substit... | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x over One minus x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_minus_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_minus_x_over_One_minus_x/Proof_1 | [
"Definite Integrals involving Logarithm Function"
] | [] | [
"Integration by Substitution/Definite Integral",
"Power Series Expansion for Logarithm of 1 - x",
"Fubini's Theorem",
"Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x",
"Gamma Function Extends Factorial"
] |
proofwiki-16238 | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x | :$\ds \int_0^1 \ln x \map \ln {1 - x} \rd x = 2 - \frac {\pi^2} 6$ | With a view to expressing the primitive in the form:
{{begin-eqn}}
{{eqn | l = \int u \frac {\d v} {\d x} \rd x
| r = u v - \int v \frac {\d u} {\d x} \rd x
| c = Integration by Parts
}}
{{end-eqn}}
let:
{{begin-eqn}}
{{eqn | l = u
| r = \ln x
| c =
}}
{{eqn | ll= \leadsto
| l = \frac {\d... | :$\ds \int_0^1 \ln x \map \ln {1 - x} \rd x = 2 - \frac {\pi^2} 6$ | With a view to expressing the [[Definition:Primitive|primitive]] in the form:
{{begin-eqn}}
{{eqn | l = \int u \frac {\d v} {\d x} \rd x
| r = u v - \int v \frac {\d u} {\d x} \rd x
| c = [[Integration by Parts]]
}}
{{end-eqn}}
let:
{{begin-eqn}}
{{eqn | l = u
| r = \ln x
| c =
}}
{{eqn | l... | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One minus x over One minus x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_minus_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_minus_x_over_One_minus_x/Proof_2 | [
"Definite Integrals involving Logarithm Function"
] | [] | [
"Definition:Primitive",
"Integration by Parts",
"Derivative of Natural Logarithm Function",
"Primitive of Power",
"Integration by Parts",
"Integration by Substitution/Definite Integral",
"Logarithm of Reciprocal",
"Integral Representation of Riemann Zeta Function in terms of Gamma Function/Corollary",... |
proofwiki-16239 | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One plus x | :$\ds \int_0^1 \ln x \map \ln {1 + x} \rd x = 2 - 2 \ln 2 - \frac {\pi^2} {12}$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \ln x \map \ln {1 + x} \rd x
| r = \int_0^1 \ln x \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n} \rd x
| c = Power Series Expansion for $\map \ln {1 + x}$
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} } n \paren {\int_0^1 x^n \ln x \rd... | :$\ds \int_0^1 \ln x \map \ln {1 + x} \rd x = 2 - 2 \ln 2 - \frac {\pi^2} {12}$ | {{begin-eqn}}
{{eqn | l = \int_0^1 \ln x \map \ln {1 + x} \rd x
| r = \int_0^1 \ln x \paren {\sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n} \rd x
| c = [[Power Series Expansion for Logarithm of 1 + x|Power Series Expansion for $\map \ln {1 + x}$]]
}}
{{eqn | r = \sum_{n \mathop = 1}^\infty \frac {\par... | Definite Integral from 0 to 1 of Logarithm of x by Logarithm of One plus x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_plus_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Logarithm_of_x_by_Logarithm_of_One_plus_x | [
"Definite Integrals involving Logarithm Function"
] | [] | [
"Power Series Expansion for Logarithm of 1 + x",
"Fubini's Theorem",
"Definite Integral from 0 to 1 of Power of x by Power of Logarithm of x",
"Gamma Function Extends Factorial",
"Newton-Mercator Series/Examples/2",
"Sum of Reciprocals of Squares Alternating in Sign"
] |
proofwiki-16240 | Set is Subset of Intersection of Supersets/Set of Sets | Let $T$ be a set.
Let $\mathbb S$ be a set of sets.
Suppose that for each $S \in \mathbb S$, $T \subseteq S$.
Then:
:$T \subseteq \ds \bigcap \mathbb S$ | Let $x \in T$.
We are given that:
:$\forall S \in \mathbb S: T \subseteq S$
Thus by definition of subset:
:$\forall S \in \mathbb S: x \in S$
Hence by definition of intersection:
:$x \in \ds \bigcap \mathbb S$
Thus by definition of subset:
:$T \subseteq \ds \bigcap \mathbb S$
{{qed}}
Category:Set is Subset of Intersec... | Let $T$ be a [[Definition:Set|set]].
Let $\mathbb S$ be a [[Definition:Set of Sets|set of sets]].
Suppose that for each $S \in \mathbb S$, $T \subseteq S$.
Then:
:$T \subseteq \ds \bigcap \mathbb S$ | Let $x \in T$.
We are given that:
:$\forall S \in \mathbb S: T \subseteq S$
Thus by definition of [[Definition:Subset|subset]]:
:$\forall S \in \mathbb S: x \in S$
Hence by definition of [[Definition:Intersection of Set of Sets|intersection]]:
:$x \in \ds \bigcap \mathbb S$
Thus by definition of [[Definition:Subse... | Set is Subset of Intersection of Supersets/Set of Sets | https://proofwiki.org/wiki/Set_is_Subset_of_Intersection_of_Supersets/Set_of_Sets | https://proofwiki.org/wiki/Set_is_Subset_of_Intersection_of_Supersets/Set_of_Sets | [
"Set is Subset of Intersection of Supersets"
] | [
"Definition:Set",
"Definition:Set of Sets"
] | [
"Definition:Subset",
"Definition:Set Intersection/Set of Sets",
"Definition:Subset",
"Category:Set is Subset of Intersection of Supersets"
] |
proofwiki-16241 | Power Series Expansion for Gauss Error Function | {{begin-eqn}}
{{eqn | l = \map \erf x
| r = \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }
| c =
}}
{{eqn | r = \frac 2 {\sqrt \pi} \paren {x - \frac {x^3} {3 \times 1!} + \frac {x^5} {5 \times 2!} - \frac {x^7} {7 \times 3!} + \frac {x^9} {9 \time... | {{begin-eqn}}
{{eqn | l = \map \erf x
| r = \frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u
| c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = \frac 2 {\sqrt \pi} \int_0^x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {n!} } \rd u
| c = {{Defof|Real Exponential Function}}
}}
{{eqn | ... | {{begin-eqn}}
{{eqn | l = \map \erf x
| r = \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }
| c =
}}
{{eqn | r = \frac 2 {\sqrt \pi} \paren {x - \frac {x^3} {3 \times 1!} + \frac {x^5} {5 \times 2!} - \frac {x^7} {7 \times 3!} + \frac {x^9} {9 \time... | {{begin-eqn}}
{{eqn | l = \map \erf x
| r = \frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u
| c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = \frac 2 {\sqrt \pi} \int_0^x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {n!} } \rd u
| c = {{Defof|Real Exponential Function}}
}}
{{eqn | ... | Power Series Expansion for Gauss Error Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Gauss_Error_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Gauss_Error_Function | [
"Examples of Power Series",
"Gauss Error Function"
] | [] | [
"Power Series is Termwise Integrable within Radius of Convergence",
"Primitive of Power"
] |
proofwiki-16242 | Set is Subset of Intersection of Supersets/General Result | Let $\family {S_i}_{i \mathop \in I}$ be a family of sets indexed by $I$.
Let $X$ be a set such that:
:$\forall i \in I: X \subseteq S_i$
Then:
:$\ds X \subseteq \bigcap_{i \mathop \in I} S_i$
where $\ds \bigcap_{i \mathop \in I} S_i$ is the intersection of $\family {S_i}$. | Let $X \subseteq S_i$ for all $i \in I$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = X
| c =
}}
{{eqn | ll= \leadsto
| q = \forall i \in I
| l = x
| o = \in
| r = S_i
| c = {{Defof|Subset}}
}}
{{eqn | ll= \leadsto
| q = \forall i \in I
| l = x
| o... | Let $\family {S_i}_{i \mathop \in I}$ be a [[Definition:Indexed Family of Sets|family of sets indexed by $I$]].
Let $X$ be a [[Definition:Set|set]] such that:
:$\forall i \in I: X \subseteq S_i$
Then:
:$\ds X \subseteq \bigcap_{i \mathop \in I} S_i$
where $\ds \bigcap_{i \mathop \in I} S_i$ is the [[Definition:Inter... | Let $X \subseteq S_i$ for all $i \in I$.
Then:
{{begin-eqn}}
{{eqn | l = x
| o = \in
| r = X
| c =
}}
{{eqn | ll= \leadsto
| q = \forall i \in I
| l = x
| o = \in
| r = S_i
| c = {{Defof|Subset}}
}}
{{eqn | ll= \leadsto
| q = \forall i \in I
| l = x
| ... | Set is Subset of Intersection of Supersets/General Result | https://proofwiki.org/wiki/Set_is_Subset_of_Intersection_of_Supersets/General_Result | https://proofwiki.org/wiki/Set_is_Subset_of_Intersection_of_Supersets/General_Result | [
"Set is Subset of Intersection of Supersets"
] | [
"Definition:Indexing Set/Family of Sets",
"Definition:Set",
"Definition:Set Intersection/Family of Sets"
] | [] |
proofwiki-16243 | Gauss Error Function is Odd | :$\map \erf {-x} = -\map \erf x$ | {{begin-eqn}}
{{eqn | l = \map \erf {-x}
| r = \frac 2 {\sqrt \pi} \int_0^{-x} e^{-u^2} \rd u
| c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = -\frac 2 {\sqrt \pi} \int_0^{-\paren {-x} } e^{-\paren {-u}^2} \rd u
| c = substituting $u \mapsto -u$
}}
{{eqn | r = -\frac 2 {\sqrt \pi} \int_0^x e^{-u^2}... | :$\map \erf {-x} = -\map \erf x$ | {{begin-eqn}}
{{eqn | l = \map \erf {-x}
| r = \frac 2 {\sqrt \pi} \int_0^{-x} e^{-u^2} \rd u
| c = {{Defof|Gauss Error Function}}
}}
{{eqn | r = -\frac 2 {\sqrt \pi} \int_0^{-\paren {-x} } e^{-\paren {-u}^2} \rd u
| c = [[Integration by Substitution|substituting]] $u \mapsto -u$
}}
{{eqn | r = -\frac... | Gauss Error Function is Odd | https://proofwiki.org/wiki/Gauss_Error_Function_is_Odd | https://proofwiki.org/wiki/Gauss_Error_Function_is_Odd | [
"Gauss Error Function",
"Examples of Odd Functions"
] | [] | [
"Integration by Substitution"
] |
proofwiki-16244 | Gauss Error Function of Zero | :$\map \erf 0 = 0$ | By Gauss Error Function is Odd, $\erf$ is a odd function.
Therefore, by Odd Function of Zero is Zero:
:$\map \erf 0 = 0$
{{qed}} | :$\map \erf 0 = 0$ | By [[Gauss Error Function is Odd]], $\erf$ is a [[Definition:Odd Function|odd function]].
Therefore, by [[Odd Function of Zero is Zero]]:
:$\map \erf 0 = 0$
{{qed}} | Gauss Error Function of Zero | https://proofwiki.org/wiki/Gauss_Error_Function_of_Zero | https://proofwiki.org/wiki/Gauss_Error_Function_of_Zero | [
"Gauss Error Function"
] | [] | [
"Gauss Error Function is Odd",
"Definition:Odd Function",
"Odd Function of Zero is Zero"
] |
proofwiki-16245 | Power Series Expansion for Complementary Error Function | {{begin-eqn}}
{{eqn | l = \map \erfc x
| r = 1 - \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }
| c =
}}
{{eqn | r = 1 - \frac 2 {\sqrt \pi} \paren {x - \frac {x^3} {3 \times 1!} + \frac {x^5} {5 \times 2!} - \frac {x^7} {7 \times 3!} + \frac {x^9}... | {{begin-eqn}}
{{eqn | l = \map \erfc x
| r = 1 - \map \erf x
| c = {{Defof|Complementary Error Function}}
}}
{{eqn | r = 1 - \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }
| c = Power Series Expansion for Gauss Error Function
}}
{{end-eqn}}
{{... | {{begin-eqn}}
{{eqn | l = \map \erfc x
| r = 1 - \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }
| c =
}}
{{eqn | r = 1 - \frac 2 {\sqrt \pi} \paren {x - \frac {x^3} {3 \times 1!} + \frac {x^5} {5 \times 2!} - \frac {x^7} {7 \times 3!} + \frac {x^9}... | {{begin-eqn}}
{{eqn | l = \map \erfc x
| r = 1 - \map \erf x
| c = {{Defof|Complementary Error Function}}
}}
{{eqn | r = 1 - \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }
| c = [[Power Series Expansion for Gauss Error Function]]
}}
{{end-eqn}... | Power Series Expansion for Complementary Error Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Complementary_Error_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Complementary_Error_Function | [
"Complementary Error Function",
"Examples of Power Series"
] | [] | [
"Power Series Expansion for Gauss Error Function"
] |
proofwiki-16246 | P-adic Expansion is a Cauchy Sequence in P-adic Norm | Let $p$ be a prime number.
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals numbers $\Q$.
Let $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be a $p$-adic expansion.
Then the sequence of partial sums of the series:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$
is a Cauchy sequence in the valued field $\struct{\... | Let $\sequence {s_N}$ be the sequence of partial sums defined by:
:$\forall N \in \Z_{\ge m}: s_N = \ds \sum_{n \mathop = m}^N d_n p^n$
From Sequence of Consecutive Integers Modulo Power of p is Cauchy in P-adic Norm:
:the sequence $\sequence {s_N}$ is a Cauchy sequence if:
::$\forall N \in \Z_{\ge m}: s_{N + 1} \equiv... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\norm {\,\cdot\,}_p$ be the [[Definition:P-adic Norm|$p$-adic norm]] on the [[Definition:Rational Numbers|rationals numbers $\Q$]].
Let $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be a [[Definition:P-adic Expansion|$p$-adic expansion]].
Then the [[Definition... | Let $\sequence {s_N}$ be the [[Definition:Sequence of Partial Sums|sequence of partial sums]] defined by:
:$\forall N \in \Z_{\ge m}: s_N = \ds \sum_{n \mathop = m}^N d_n p^n$
From [[Sequence of Consecutive Integers Modulo Power of p is Cauchy in P-adic Norm]]:
:the [[Definition:Sequence of Partial Sums|sequence]] $\... | P-adic Expansion is a Cauchy Sequence in P-adic Norm | https://proofwiki.org/wiki/P-adic_Expansion_is_a_Cauchy_Sequence_in_P-adic_Norm | https://proofwiki.org/wiki/P-adic_Expansion_is_a_Cauchy_Sequence_in_P-adic_Norm | [
"P-adic Number Theory",
"P-adic Expansion is a Cauchy Sequence in P-adic Norm"
] | [
"Definition:Prime Number",
"Definition:P-adic Norm",
"Definition:Rational Number",
"Definition:P-adic Expansion",
"Definition:Series/Sequence of Partial Sums",
"Definition:Series",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Valued Field",
"P-adic Expansion is a Cauchy Sequence in... | [
"Definition:Series/Sequence of Partial Sums",
"Characterisation of Cauchy Sequence in Non-Archimedean Norm/Corollary 1",
"Definition:Series/Sequence of Partial Sums",
"Definition:Cauchy Sequence/Normed Division Ring"
] |
proofwiki-16247 | P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. | From P-adic Expansion is a Cauchy Sequence in P-adic Norm, the sequence of partial sums of the series:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$
is a Cauchy sequence in the rationals $\Q$ with the $p$-adic norm.
From Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers, the sequence of partial sums of the se... | Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]]. | From [[P-adic Expansion is a Cauchy Sequence in P-adic Norm]], the [[Definition:Sequence of Partial Sums|sequence of partial sums]] of the [[Definition:Series|series]]:
:$\ds \sum_{n \mathop = m}^\infty d_n p^n$
is a [[Definition:Cauchy Sequence in Normed Division Ring|Cauchy sequence]] in the [[Definition:Rational Num... | P-adic Expansion is a Cauchy Sequence in P-adic Norm/Converges to P-adic Number | https://proofwiki.org/wiki/P-adic_Expansion_is_a_Cauchy_Sequence_in_P-adic_Norm/Converges_to_P-adic_Number | https://proofwiki.org/wiki/P-adic_Expansion_is_a_Cauchy_Sequence_in_P-adic_Norm/Converges_to_P-adic_Number | [
"P-adic Expansion is a Cauchy Sequence in P-adic Norm"
] | [
"Definition:Valued Field of P-adic Numbers"
] | [
"P-adic Expansion is a Cauchy Sequence in P-adic Norm",
"Definition:Series/Sequence of Partial Sums",
"Definition:Series",
"Definition:Cauchy Sequence/Normed Division Ring",
"Definition:Rational Number",
"Definition:P-adic Norm",
"Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers",
"Defin... |
proofwiki-16248 | Characterization of Exponential Integral Function/Formulation 1 | Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\ds \map \Ei x = -\gamma - \ln x + \int_0^x \frac {1 - e^{-u} } u \rd u$ | We have, by Derivative of $e^{a x}$:
:$\map {\dfrac \d {\d u} } {1 - e^{-u} } = e^{-u}$
By Primitive of Reciprocal:
:$\ds \int \frac {\d u} u = \ln u + C$
So:
{{begin-eqn}}
{{eqn | l = \int_0^x \frac {1 - e^{-u} } u \rd u
| r = \bigintlimits {\paren {1 - e^{-u} } \ln u} 0 x - \int_0^x e^{-u} \ln u \rd u
|... | Let $\Ei: \R_{>0} \to \R$ denote the [[Definition:Exponential Integral Function/Formulation 1|exponential integral function]]:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\ds \map \Ei x = -\gamma - \ln x + \int_0^x \frac {1 - e^{-u} } u \rd u$ | We have, by [[Derivative of Exponential of a x|Derivative of $e^{a x}$]]:
:$\map {\dfrac \d {\d u} } {1 - e^{-u} } = e^{-u}$
By [[Primitive of Reciprocal]]:
:$\ds \int \frac {\d u} u = \ln u + C$
So:
{{begin-eqn}}
{{eqn | l = \int_0^x \frac {1 - e^{-u} } u \rd u
| r = \bigintlimits {\paren {1 - e^{-u} } \l... | Characterization of Exponential Integral Function/Formulation 1 | https://proofwiki.org/wiki/Characterization_of_Exponential_Integral_Function/Formulation_1 | https://proofwiki.org/wiki/Characterization_of_Exponential_Integral_Function/Formulation_1 | [
"Characterization of Exponential Integral Function"
] | [
"Definition:Exponential Integral Function/Formulation 1"
] | [
"Derivative of Exponential Function/Corollary 1",
"Primitive of Reciprocal",
"Integration by Parts",
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Combination Theorem for Limits of Functions/Real/Product Rule",
"Derivative of Exponential Function/Corollary 1",
"Limit of Power of x ... |
proofwiki-16249 | Complementary Error Function of Zero | :$\map \erfc 0 = 1$ | {{begin-eqn}}
{{eqn | l = \map \erfc 0
| r = 1 - \map \erf 0
| c = {{Defof|Complementary Error Function}}
}}
{{eqn | r = 1
| c = Gauss Error Function of Zero
}}
{{end-eqn}}
{{qed}} | :$\map \erfc 0 = 1$ | {{begin-eqn}}
{{eqn | l = \map \erfc 0
| r = 1 - \map \erf 0
| c = {{Defof|Complementary Error Function}}
}}
{{eqn | r = 1
| c = [[Gauss Error Function of Zero]]
}}
{{end-eqn}}
{{qed}} | Complementary Error Function of Zero | https://proofwiki.org/wiki/Complementary_Error_Function_of_Zero | https://proofwiki.org/wiki/Complementary_Error_Function_of_Zero | [
"Complementary Error Function"
] | [] | [
"Gauss Error Function of Zero"
] |
proofwiki-16250 | Limit to Infinity of Complementary Error Function | :$\ds \lim_{x \mathop \to \infty} \map \erfc x = 0$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \map \erfc x
| r = \lim_{x \mathop \to \infty} \paren {1 - \map \erf x}
| c = {{Defof|Complementary Error Function}}
}}
{{eqn | r = 1 - \lim_{x \mathop \to \infty} \map \erf x
| c = Sum Rule for Limits of Real Functions
}}
{{eqn | r = 1 - 1
|... | :$\ds \lim_{x \mathop \to \infty} \map \erfc x = 0$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \map \erfc x
| r = \lim_{x \mathop \to \infty} \paren {1 - \map \erf x}
| c = {{Defof|Complementary Error Function}}
}}
{{eqn | r = 1 - \lim_{x \mathop \to \infty} \map \erf x
| c = [[Sum Rule for Limits of Real Functions]]
}}
{{eqn | r = 1 - 1
... | Limit to Infinity of Complementary Error Function | https://proofwiki.org/wiki/Limit_to_Infinity_of_Complementary_Error_Function | https://proofwiki.org/wiki/Limit_to_Infinity_of_Complementary_Error_Function | [
"Complementary Error Function"
] | [] | [
"Combination Theorem for Limits of Functions/Real/Sum Rule",
"Limit to Infinity of Gauss Error Function"
] |
proofwiki-16251 | Membership Relation is Not Reflexive | Let $\Bbb S$ be a set of sets in the context of pure set theory
Let $\RR$ denote the membership relation on $\Bbb S$:
:$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$
$\RR$ is not in general a reflexive relation. | In the extreme pathological edge case:
:$S = \set S$
it is seen that:
:$S \in S$
and so:
:$\forall x \in S: x \in x$
demonstrating that $\RR$ is reflexive in this specific case.
However, in this case $\set S$ is a set on which the Axiom of Foundation does not apply.
This is seen in Set is Not Element of Itself.
Hence t... | Let $\Bbb S$ be a [[Definition:Set of Sets|set of sets]] in the context of [[Definition:Pure Set Theory|pure set theory]]
Let $\RR$ denote the [[Definition:Membership Relation|membership relation]] on $\Bbb S$:
:$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$
$\RR$ is not in gene... | In the extreme pathological edge case:
:$S = \set S$
it is seen that:
:$S \in S$
and so:
:$\forall x \in S: x \in x$
demonstrating that $\RR$ is [[Definition:Reflexive Relation|reflexive]] in this specific case.
However, in this case $\set S$ is a [[Definition:Set|set]] on which the [[Axiom:Axiom of Foundation|Axio... | Membership Relation is Not Reflexive | https://proofwiki.org/wiki/Membership_Relation_is_Not_Reflexive | https://proofwiki.org/wiki/Membership_Relation_is_Not_Reflexive | [
"Membership Relation"
] | [
"Definition:Set of Sets",
"Definition:Pure Set Theory",
"Definition:Membership Relation",
"Definition:Reflexive Relation"
] | [
"Definition:Reflexive Relation",
"Definition:Set",
"Axiom:Axiom of Foundation",
"Set is Not Element of Itself",
"Definition:Set",
"Definition:Zermelo-Fraenkel Set Theory",
"Definition:Set",
"Definition:Reflexive Relation"
] |
proofwiki-16252 | Membership Relation is Not Symmetric | Let $\Bbb S$ be a set of sets in the context of pure set theory
Let $\RR$ denote the membership relation on $\Bbb S$:
:$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$
$\RR$ is not in general a symmetric relation. | In the extreme pathological edge case:
:$S = \set S$
it is seen that:
:$S \in S$
and so:
:$\forall x \in S: \tuple {a, b} \in \RR \implies \tuple {b, a} \in \RR$
demonstrating that $\RR$ is symmetric in this specific case.
However, in this case $\set S$ is a set on which the Axiom of Foundation does not apply.
This is ... | Let $\Bbb S$ be a [[Definition:Set of Sets|set of sets]] in the context of [[Definition:Pure Set Theory|pure set theory]]
Let $\RR$ denote the [[Definition:Membership Relation|membership relation]] on $\Bbb S$:
:$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$
$\RR$ is not in gene... | In the extreme pathological edge case:
:$S = \set S$
it is seen that:
:$S \in S$
and so:
:$\forall x \in S: \tuple {a, b} \in \RR \implies \tuple {b, a} \in \RR$
demonstrating that $\RR$ is [[Definition:Symmetric Relation|symmetric]] in this specific case.
However, in this case $\set S$ is a [[Definition:Set|set]] ... | Membership Relation is Not Symmetric | https://proofwiki.org/wiki/Membership_Relation_is_Not_Symmetric | https://proofwiki.org/wiki/Membership_Relation_is_Not_Symmetric | [
"Membership Relation"
] | [
"Definition:Set of Sets",
"Definition:Pure Set Theory",
"Definition:Membership Relation",
"Definition:Symmetric Relation"
] | [
"Definition:Symmetric Relation",
"Definition:Set",
"Axiom:Axiom of Foundation",
"Set is Not Element of Itself",
"Definition:Set",
"Definition:Zermelo-Fraenkel Set Theory",
"Definition:Set",
"Definition:Symmetric Relation"
] |
proofwiki-16253 | Membership Relation is Antisymmetric | Let $\Bbb S$ be a set of sets in the context of pure set theory
Let $\RR$ denote the membership relation on $\Bbb S$:
:$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$
$\RR$ is an antisymmetric relation. | {{ProofWanted|Applies in the case where the Axiom of Foundation applies, but I'm not so sure about non-standard sets.}} | Let $\Bbb S$ be a [[Definition:Set of Sets|set of sets]] in the context of [[Definition:Pure Set Theory|pure set theory]]
Let $\RR$ denote the [[Definition:Membership Relation|membership relation]] on $\Bbb S$:
:$\forall \tuple {a, b} \in \Bbb S \times \Bbb S: \tuple {a, b} \in \RR \iff a \in b$
$\RR$ is an [[Defini... | {{ProofWanted|Applies in the case where the [[Axiom:Axiom of Foundation|Axiom of Foundation]] applies, but I'm not so sure about non-standard sets.}} | Membership Relation is Antisymmetric | https://proofwiki.org/wiki/Membership_Relation_is_Antisymmetric | https://proofwiki.org/wiki/Membership_Relation_is_Antisymmetric | [
"Membership Relation"
] | [
"Definition:Set of Sets",
"Definition:Pure Set Theory",
"Definition:Membership Relation",
"Definition:Antisymmetric Relation"
] | [
"Axiom:Axiom of Foundation"
] |
proofwiki-16254 | Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Q$ denote the rational numbers identified as a dense subfield of $\struct {\Q_p, \norm {\,\cdot\,}_p}$.
Let $\norm {\,\cdot\,} ^\Q _p$ denote the $p$-adic norm on the rational numbers.
Let $\sequence{x_n}$ be a sequence ... | From Rational Numbers are Dense Subfield of P-adic Numbers:
:the $p$-adic norm $\norm {\,\cdot\,}_p$ on the $p$-adic numbers is an extension of the $p$-adic norm $\norm {\,\cdot\,} ^\Q _p$ on the rational numbers.
That is,
:$\norm {\,\cdot\,}_p \restriction_\Q \mathop = \norm {\,\cdot\,} ^\Q _p$
The result follows from... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Q$ denote the [[Definition:Rational Number|rational numbers]] identified as a [[Definition:Everywhere Dense|dense]] [[Definition:Subfield|subfi... | From [[Rational Numbers are Dense Subfield of P-adic Numbers]]:
:the [[Definition:P-adic Norm on P-adic Numbers|$p$-adic norm]] $\norm {\,\cdot\,}_p$ on the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]] is an [[Definition:Extension of Mapping|extension]] of the [[Definition:P-adic Norm|$p$-adic norm]] ... | Sequence is Cauchy in P-adic Norm iff Cauchy in P-adic Numbers | https://proofwiki.org/wiki/Sequence_is_Cauchy_in_P-adic_Norm_iff_Cauchy_in_P-adic_Numbers | https://proofwiki.org/wiki/Sequence_is_Cauchy_in_P-adic_Norm_iff_Cauchy_in_P-adic_Numbers | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:Rational Number",
"Definition:Everywhere Dense",
"Definition:Subfield",
"Definition:P-adic Norm/Rational Numbers",
"Definition:Rational Number",
"Definition:Sequence",
"Definition:Rational Number",
"Definition:Cauc... | [
"Rational Numbers are Dense Subfield of P-adic Numbers",
"Definition:P-adic Norm/P-adic Numbers",
"Definition:Valued Field of P-adic Numbers",
"Definition:Extension of Mapping",
"Definition:P-adic Norm",
"Definition:Rational Number",
"Cauchy Sequence of Subring iff Cauchy Sequence of Normed Division Rin... |
proofwiki-16255 | Definite Integral to Infinity of Sine x over Root x | :$\ds \int_0^\infty \frac {\sin x} {\sqrt x} \rd x = \sqrt {\frac \pi 2}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sin x} {\sqrt x} \rd x
| r = \int_0^\infty 2 u \frac {\sin u^2} {\sqrt {u^2} } \rd u
| c = substituting $x = u^2$
}}
{{eqn | r = 2 \int_0^\infty \sin u^2 \rd u
}}
{{eqn | r = \sqrt {\frac \pi 2}
| c = Definite Integral to Infinity of $\sin a x^2$
}}
{{end-eqn}}
{{qed}} | :$\ds \int_0^\infty \frac {\sin x} {\sqrt x} \rd x = \sqrt {\frac \pi 2}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sin x} {\sqrt x} \rd x
| r = \int_0^\infty 2 u \frac {\sin u^2} {\sqrt {u^2} } \rd u
| c = [[Integration by Substitution|substituting]] $x = u^2$
}}
{{eqn | r = 2 \int_0^\infty \sin u^2 \rd u
}}
{{eqn | r = \sqrt {\frac \pi 2}
| c = [[Definite Integral to Infinity of S... | Definite Integral to Infinity of Sine x over Root x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Sine_x_over_Root_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Sine_x_over_Root_x | [
"Definite Integrals involving Sine Function"
] | [] | [
"Integration by Substitution",
"Definite Integral to Infinity of Sine of a x^2"
] |
proofwiki-16256 | Digamma Function of One Half | :$\map \psi {\dfrac 1 2} = -\gamma - 2 \ln 2$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 2}
| r = -\gamma - \ln 4 - \frac \pi 2 \map \cot {\frac 1 2 \pi} + 2 \sum_{n \mathop = 1}^0 \map \cos {\frac {2 \pi n} 2} \map \ln {\map \sin {\frac {\pi n} 2} }
| c = Gauss's Digamma Theorem
}}
{{eqn | r = -\gamma - \ln 4
| c = Cotangent of Right Angle, no... | :$\map \psi {\dfrac 1 2} = -\gamma - 2 \ln 2$ | {{begin-eqn}}
{{eqn | l = \map \psi {\frac 1 2}
| r = -\gamma - \ln 4 - \frac \pi 2 \map \cot {\frac 1 2 \pi} + 2 \sum_{n \mathop = 1}^0 \map \cos {\frac {2 \pi n} 2} \map \ln {\map \sin {\frac {\pi n} 2} }
| c = [[Gauss's Digamma Theorem]]
}}
{{eqn | r = -\gamma - \ln 4
| c = [[Cotangent of Right Ang... | Digamma Function of One Half/Proof 1 | https://proofwiki.org/wiki/Digamma_Function_of_One_Half | https://proofwiki.org/wiki/Digamma_Function_of_One_Half/Proof_1 | [
"Digamma Function of One Half",
"Examples of Digamma Function",
"Euler-Mascheroni Constant"
] | [] | [
"Gauss's Digamma Theorem",
"Cotangent of Right Angle",
"Definition:Summation/Vacuous Summation",
"Logarithm of Power"
] |
proofwiki-16257 | Definite Integral to Infinity of Exponential of -x^2 by Logarithm of x | :$\ds \int_0^\infty e^{-x^2} \ln x \rd x = -\frac {\sqrt \pi} 4 \paren {\gamma + 2 \ln 2}$ | Consider the integral:
:$\ds \int_0^\infty x^t e^{-x^2} \rd x$
for positive real parameter $t$.
Using Definite Integral to Infinity of $x^m e^{-a x^2}$, we have:
:$\ds \int_0^\infty x^t e^{-x^2} \rd x = \frac 1 2 \map \Gamma {\frac {1 + t} 2}$
Differentiating the {{LHS}} {{WRT|Differentiation}} $t$ we have:
{{begin-... | :$\ds \int_0^\infty e^{-x^2} \ln x \rd x = -\frac {\sqrt \pi} 4 \paren {\gamma + 2 \ln 2}$ | Consider the integral:
:$\ds \int_0^\infty x^t e^{-x^2} \rd x$
for [[Definition:Positive Real Number|positive real parameter]] $t$.
Using [[Definite Integral to Infinity of Power of x by Exponential of -a x^2|Definite Integral to Infinity of $x^m e^{-a x^2}$]], we have:
:$\ds \int_0^\infty x^t e^{-x^2} \rd x = \f... | Definite Integral to Infinity of Exponential of -x^2 by Logarithm of x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-x^2_by_Logarithm_of_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-x^2_by_Logarithm_of_x | [
"Definite Integrals involving Exponential Function",
"Definite Integrals involving Logarithm Function",
"Euler-Mascheroni Constant"
] | [] | [
"Definition:Positive/Real Number",
"Definite Integral to Infinity of Power of x by Exponential of -a x^2",
"Definition:Differentiation",
"Definite Integral of Partial Derivative",
"Derivative of General Exponential Function",
"Definition:Differentiation",
"Derivative of Composite Function",
"Gamma Fun... |
proofwiki-16258 | Definite Integral to Infinity of Cosine x over Root x | :$\ds \int_0^\infty \frac {\cos x} {\sqrt x} \rd x = \sqrt {\frac \pi 2}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\cos x} {\sqrt x} \rd x
| r = \int_0^\infty 2 u \frac {\cos u^2} {\sqrt {u^2} } \rd u
| c = substituting $x = u^2$
}}
{{eqn | r = 2 \int_0^\infty \cos u^2 \rd u
}}
{{eqn | r = \sqrt {\frac \pi 2}
| c = Definite Integral to Infinity of $\cos a x^2$
}}
{{end-eqn}}
{{qed}} | :$\ds \int_0^\infty \frac {\cos x} {\sqrt x} \rd x = \sqrt {\frac \pi 2}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\cos x} {\sqrt x} \rd x
| r = \int_0^\infty 2 u \frac {\cos u^2} {\sqrt {u^2} } \rd u
| c = [[Integration by Substitution|substituting]] $x = u^2$
}}
{{eqn | r = 2 \int_0^\infty \cos u^2 \rd u
}}
{{eqn | r = \sqrt {\frac \pi 2}
| c = [[Definite Integral to Infinity of C... | Definite Integral to Infinity of Cosine x over Root x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cosine_x_over_Root_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cosine_x_over_Root_x | [
"Definite Integrals involving Cosine Function"
] | [] | [
"Integration by Substitution",
"Definite Integral to Infinity of Cosine of a x^2"
] |
proofwiki-16259 | Definite Integral to Infinity of Cube of Sine x over x Cubed | :$\ds \int_0^\infty \frac {\sin^3 x} {x^3} \rd x = \frac {3 \pi} 8$ | Let:
:$\ds \map I \alpha = \int_0^\infty \frac {\map {\sin^3} {\alpha x} } {x^3} \rd x$
for positive real parameter $\alpha$.
We have:
{{begin-eqn}}
{{eqn | l = \map I 0
| r = \int_0^\infty \frac {\map {\sin^3} {0 x} } {x^3} \rd x
}}
{{eqn | r = \int_0^\infty \frac 0 {x^3} \rd x
| c = Sine of Zero is Zero
}}
{{eqn ... | :$\ds \int_0^\infty \frac {\sin^3 x} {x^3} \rd x = \frac {3 \pi} 8$ | Let:
:$\ds \map I \alpha = \int_0^\infty \frac {\map {\sin^3} {\alpha x} } {x^3} \rd x$
for [[Definition:Positive Real Number|positive real parameter]] $\alpha$.
We have:
{{begin-eqn}}
{{eqn | l = \map I 0
| r = \int_0^\infty \frac {\map {\sin^3} {0 x} } {x^3} \rd x
}}
{{eqn | r = \int_0^\infty \frac 0 {x^3} \rd... | Definite Integral to Infinity of Cube of Sine x over x Cubed | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cube_of_Sine_x_over_x_Cubed | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cube_of_Sine_x_over_x_Cubed | [
"Definite Integrals involving Sine Function"
] | [] | [
"Definition:Positive/Real Number",
"Sine of Zero is Zero",
"Definite Integral of Partial Derivative",
"Power Reduction Formulas/Sine Cubed",
"Derivative of Cosine Function",
"Derivative of Composite Function",
"Definite Integral to Infinity of Cosine p x minus Cosine q x over x Squared",
"Primitive of... |
proofwiki-16260 | Equivalence of Definitions of Composition of Mappings | {{TFAE|def = Composition of Mappings}}
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$. | Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that:
:$\Dom {f_2} = \Cdm {f_1}$ | {{TFAE|def = Composition of Mappings}}
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be [[Definition:Mapping|mappings]] such that the [[Definition:Domain of Mapping|domain]] of $f_2$ is the same set as the [[Definition:Codomain of Mapping|codomain]] of $f_1$. | Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be [[Definition:Mapping|mappings]] such that:
:$\Dom {f_2} = \Cdm {f_1}$ | Equivalence of Definitions of Composition of Mappings | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Composition_of_Mappings | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Composition_of_Mappings | [
"Composite Mappings"
] | [
"Definition:Mapping",
"Definition:Domain (Set Theory)/Mapping",
"Definition:Codomain (Set Theory)/Mapping"
] | [
"Definition:Mapping",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Mapping",
"Definition:Mapping"
] |
proofwiki-16261 | Power Series Expansion for Exponential Integral Function/Formulation 1 | Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
{{begin-eqn}}
{{eqn | l = \map \Ei x
| r = -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {x^n} {n \times n!}
| c =
}}
... | {{begin-eqn}}
{{eqn | l = \map \Ei x
| r = -\gamma - \ln x + \int_0^x \frac {1 - e^{-u} } u \rd u
| c = Characterization of Exponential Integral Function
}}
{{eqn | r = -\gamma - \ln x + \int_0^x \frac 1 u \paren {1 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^n} {n!} } \rd u
| c = {{Defof|Rea... | Let $\Ei: \R_{>0} \to \R$ denote the [[Definition:Exponential Integral Function/Formulation 1|exponential integral function]]:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
{{begin-eqn}}
{{eqn | l = \map \Ei x
| r = -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \... | {{begin-eqn}}
{{eqn | l = \map \Ei x
| r = -\gamma - \ln x + \int_0^x \frac {1 - e^{-u} } u \rd u
| c = [[Characterization of Exponential Integral Function/Formulation 1|Characterization of Exponential Integral Function]]
}}
{{eqn | r = -\gamma - \ln x + \int_0^x \frac 1 u \paren {1 - \sum_{n \mathop = 0}^\... | Power Series Expansion for Exponential Integral Function/Formulation 1 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_Integral_Function/Formulation_1 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_Integral_Function/Formulation_1 | [
"Power Series Expansion for Exponential Integral Function"
] | [
"Definition:Exponential Integral Function/Formulation 1"
] | [
"Characterization of Exponential Integral Function/Formulation 1",
"Power Series is Termwise Integrable within Radius of Convergence",
"Primitive of Power"
] |
proofwiki-16262 | P-adic Numbers are Generated Ring Extension of P-adic Integers | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Then:
:$Q_p = \Z_p \sqbrk {1 / p}$
where $\Z_p \sqbrk {1 / p}$ denotes the ring extension generated by $1 / p$. | Let $a \in \Q_p$.
From P-adic Number times Integer Power of p is P-adic Integer, there exists $n \in \N$ such that $p^n a \in \Z_p$.
Since $n \in \N$ and $p^n a \in \Z_p$, let $\map f X \in \Z_p \sqbrk X$ be the polynomial:
:$\paren {p^n a} X^n$
Then:
:$\map f {1 / p} = \paren {p^n a} \paren {1 / p}^n = a$.
Hence:
:$a ... | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Then:
:$Q_p = \Z_p \sqbrk {1 / p}$
where $\Z_p \sqbrk {1 / p}$ denotes the [[Defi... | Let $a \in \Q_p$.
From [[P-adic Number times Integer Power of p is P-adic Integer]], there exists $n \in \N$ such that $p^n a \in \Z_p$.
Since $n \in \N$ and $p^n a \in \Z_p$, let $\map f X \in \Z_p \sqbrk X$ be the [[Definition:Polynomial (Abstract Algebra)|polynomial]]:
:$\paren {p^n a} X^n$
Then:
:$\map f {1 / p}... | P-adic Numbers are Generated Ring Extension of P-adic Integers | https://proofwiki.org/wiki/P-adic_Numbers_are_Generated_Ring_Extension_of_P-adic_Integers | https://proofwiki.org/wiki/P-adic_Numbers_are_Generated_Ring_Extension_of_P-adic_Integers | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer",
"Definition:Generated Ring Extension"
] | [
"P-adic Number times Integer Power of p is P-adic Integer",
"Definition:Polynomial over Ring",
"Definition:Generated Ring Extension"
] |
proofwiki-16263 | Asymptotic Expansion for Exponential Integral Function/Formulation 1 | Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
{{begin-eqn}}
{{eqn | l = \map \Ei x
| o = \sim
| r = \frac {e^{-x} } x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {n!} {x^n}
| c =
}... | {{begin-eqn}}
{{eqn | l = \map \Ei x
| r = \int_x^\infty \frac {e^{-t} } t \rd t
| c = {{Defof|Exponential Integral Function/Formulation 1|Exponential Integral Function}}
}}
{{eqn | r = \intlimits {-\frac {e^{-t} } t} {t \mathop = x} {t \mathop \to \infty} -\int_x^\infty \frac {e^{-t} } {t^2} \rd t
| ... | Let $\Ei: \R_{>0} \to \R$ denote the [[Definition:Exponential Integral Function/Formulation 1|exponential integral function]]:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
{{begin-eqn}}
{{eqn | l = \map \Ei x
| o = \sim
| r = \frac {e^{-x} } x \sum_{n \mat... | {{begin-eqn}}
{{eqn | l = \map \Ei x
| r = \int_x^\infty \frac {e^{-t} } t \rd t
| c = {{Defof|Exponential Integral Function/Formulation 1|Exponential Integral Function}}
}}
{{eqn | r = \intlimits {-\frac {e^{-t} } t} {t \mathop = x} {t \mathop \to \infty} -\int_x^\infty \frac {e^{-t} } {t^2} \rd t
| ... | Asymptotic Expansion for Exponential Integral Function/Formulation 1 | https://proofwiki.org/wiki/Asymptotic_Expansion_for_Exponential_Integral_Function/Formulation_1 | https://proofwiki.org/wiki/Asymptotic_Expansion_for_Exponential_Integral_Function/Formulation_1 | [
"Asymptotic Expansion for Exponential Integral Function"
] | [
"Definition:Exponential Integral Function/Formulation 1"
] | [
"Integration by Parts",
"Integration by Parts",
"Integration by Parts"
] |
proofwiki-16264 | Limit to Infinity of Exponential Integral Function | Let $\Ei: \R_{>0} \to \R$ denote the exponential integral function:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\ds \lim_{x \mathop \to \infty} \map \Ei x = 0$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \map \Ei x
| r = \lim_{x \mathop \to \infty} \int_x^\infty \frac {e^{-u} } u \rd u
| c = {{Defof|Exponential Integral Function/Formulation 1|Exponential Integral Function}}
}}
{{eqn | r = \lim_{x \mathop \to \infty} \int_1^\infty \frac {e^{-x t} } {x t} ... | Let $\Ei: \R_{>0} \to \R$ denote the [[Definition:Exponential Integral Function/Formulation 1|exponential integral function]]:
:$\map \Ei x = \ds \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {e^{-t} } t \rd t$
Then:
:$\ds \lim_{x \mathop \to \infty} \map \Ei x = 0$ | {{begin-eqn}}
{{eqn | l = \lim_{x \mathop \to \infty} \map \Ei x
| r = \lim_{x \mathop \to \infty} \int_x^\infty \frac {e^{-u} } u \rd u
| c = {{Defof|Exponential Integral Function/Formulation 1|Exponential Integral Function}}
}}
{{eqn | r = \lim_{x \mathop \to \infty} \int_1^\infty \frac {e^{-x t} } {x t} ... | Limit to Infinity of Exponential Integral Function | https://proofwiki.org/wiki/Limit_to_Infinity_of_Exponential_Integral_Function | https://proofwiki.org/wiki/Limit_to_Infinity_of_Exponential_Integral_Function | [
"Exponential Integral Function"
] | [
"Definition:Exponential Integral Function/Formulation 1"
] | [
"Integration by Substitution",
"Lebesgue's Dominated Convergence Theorem",
"Exponential Tends to Zero and Infinity"
] |
proofwiki-16265 | Definite Integral to Infinity of Power of x over Hyperbolic Sine of a x | :$\ds \int_0^\infty \frac {x^n} {\sinh a x} \rd x = \frac {2^{n + 1} - 1} {2^n a^{n + 1} } \map \Gamma {n + 1} \map \zeta {n + 1}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {x^n} {\sinh a x} \rd x
| r = \frac 1 a \int_0^\infty \frac {\paren {\frac u a}^n} {\sinh u} \rd u
| c = substituting $u = a x$
}}
{{eqn | r = \frac 2 {a^{n + 1} } \int_0^\infty \frac {u^n e^{-u} } {1 - e^{-2 u} } \rd u
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \fra... | :$\ds \int_0^\infty \frac {x^n} {\sinh a x} \rd x = \frac {2^{n + 1} - 1} {2^n a^{n + 1} } \map \Gamma {n + 1} \map \zeta {n + 1}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {x^n} {\sinh a x} \rd x
| r = \frac 1 a \int_0^\infty \frac {\paren {\frac u a}^n} {\sinh u} \rd u
| c = [[Integration by Substitution|substituting]] $u = a x$
}}
{{eqn | r = \frac 2 {a^{n + 1} } \int_0^\infty \frac {u^n e^{-u} } {1 - e^{-2 u} } \rd u
| c = {{Defof|Hyper... | Definite Integral to Infinity of Power of x over Hyperbolic Sine of a x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_over_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Power_of_x_over_Hyperbolic_Sine_of_a_x | [
"Definite Integrals involving Hyperbolic Sine Function",
"Gamma Function",
"Riemann Zeta Function"
] | [] | [
"Integration by Substitution",
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Laplace Transform of Real Power"
] |
proofwiki-16266 | Definite Integral to Infinity of x over Hyperbolic Sine of a x | :$\ds \int_0^\infty \frac x {\sinh a x} \rd x = \frac {\pi^2} {4 a^2}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac x {\sinh a x} \rd x
| r = \frac {2^2 - 1} {2 a^2} \map \Gamma 2 \map \zeta 2
| c = Definite Integral to Infinity of $\dfrac {x^n} {\sinh a x}$
}}
{{eqn | r = \frac 3 {2 a^2} \times 1! \times \frac {\pi^2} 6
| c = Gamma Function Extends Factorial, Basel Problem
}}
{{eqn |... | :$\ds \int_0^\infty \frac x {\sinh a x} \rd x = \frac {\pi^2} {4 a^2}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac x {\sinh a x} \rd x
| r = \frac {2^2 - 1} {2 a^2} \map \Gamma 2 \map \zeta 2
| c = [[Definite Integral to Infinity of Power of x over Hyperbolic Sine of a x|Definite Integral to Infinity of $\dfrac {x^n} {\sinh a x}$]]
}}
{{eqn | r = \frac 3 {2 a^2} \times 1! \times \frac... | Definite Integral to Infinity of x over Hyperbolic Sine of a x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_x_over_Hyperbolic_Sine_of_a_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_x_over_Hyperbolic_Sine_of_a_x | [
"Definite Integrals involving Hyperbolic Sine Function"
] | [] | [
"Definite Integral to Infinity of Power of x over Hyperbolic Sine of a x",
"Gamma Function Extends Factorial",
"Basel Problem"
] |
proofwiki-16267 | Fourier Series/Identity Function over Symmetric Range | Let $\lambda \in \R_{>0}$ be a strictly positive real number.
Let $\map f x: \openint {-\lambda} \lambda \to \R$ be the identity function on the open real interval $\openint {-\lambda} \lambda$:
:$\forall x \in \openint {-\lambda} \lambda: \map f x = x$
The Fourier series of $f$ over $\openint {-\lambda} \lambda$ can b... | From Identity Function is Odd Function, $\map f x$ is a odd function.
By Fourier Series for Odd Function over Symmetric Range, we have:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$
where:
{{begin-eqn}}
{{eqn | l = b_n
| r = \frac 2 \lambda \int_0^\pi \map f x \sin \frac {n \pi... | Let $\lambda \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $\map f x: \openint {-\lambda} \lambda \to \R$ be the [[Definition:Identity Function|identity function]] on the [[Definition:Open Real Interval|open real interval]] $\openint {-\lambda} \lambda$:
:$\forall x ... | From [[Identity Function is Odd Function]], $\map f x$ is a [[Definition:Odd Function|odd function]].
By [[Fourier Series for Odd Function over Symmetric Range]], we have:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$
where:
{{begin-eqn}}
{{eqn | l = b_n
| r = \frac 2 \lamb... | Fourier Series/Identity Function over Symmetric Range | https://proofwiki.org/wiki/Fourier_Series/Identity_Function_over_Symmetric_Range | https://proofwiki.org/wiki/Fourier_Series/Identity_Function_over_Symmetric_Range | [
"Identity Mappings",
"Fourier Series for Identity Function"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Identity Mapping",
"Definition:Real Interval/Open",
"Definition:Fourier Series"
] | [
"Identity Function is Odd Function",
"Definition:Odd Function",
"Fourier Series for Odd Function over Symmetric Range",
"Half-Range Fourier Series/Identity Function/Sine",
"Category:Identity Mappings",
"Category:Fourier Series for Identity Function"
] |
proofwiki-16268 | Restriction of Mapping is its Intersection with Cartesian Product of Subset with Image | Let $f: S \to T$ be a mapping.
Let $X \subseteq S$.
Let $f {\restriction_X}$ be the restriction of $f$ to $X$.
Then:
:$f {\restriction_X} = f \cap \paren {X \times \Img f}$
where:
:$\Img f$ denotes the image of $f$, defined as:
::$\Img f = \set {t \in T: \exists s \in S: t = \map f s}$
:$X \times \Img f$ denotes the ca... | By Restriction of Mapping is Mapping, $f {\restriction_X}: X \to T$ is a mapping.
We have:
{{begin-eqn}}
{{eqn | l = \tuple {x, y}
| o = \in
| r = f {\restriction_X}
| c =
}}
{{eqn | ll= \leadsto
| l = \tuple {x, y}
| o = \in
| r = f
| c = {{Defof|Restriction of Mapping}}
}}
{... | Let $f: S \to T$ be a [[Definition:Mapping|mapping]].
Let $X \subseteq S$.
Let $f {\restriction_X}$ be the [[Definition:Restriction of Mapping|restriction of $f$ to $X$]].
Then:
:$f {\restriction_X} = f \cap \paren {X \times \Img f}$
where:
:$\Img f$ denotes the [[Definition:Image of Mapping|image]] of $f$, defined... | By [[Restriction of Mapping is Mapping]], $f {\restriction_X}: X \to T$ is a [[Definition:Mapping|mapping]].
We have:
{{begin-eqn}}
{{eqn | l = \tuple {x, y}
| o = \in
| r = f {\restriction_X}
| c =
}}
{{eqn | ll= \leadsto
| l = \tuple {x, y}
| o = \in
| r = f
| c = {{Defof|... | Restriction of Mapping is its Intersection with Cartesian Product of Subset with Image | https://proofwiki.org/wiki/Restriction_of_Mapping_is_its_Intersection_with_Cartesian_Product_of_Subset_with_Image | https://proofwiki.org/wiki/Restriction_of_Mapping_is_its_Intersection_with_Cartesian_Product_of_Subset_with_Image | [
"Restrictions"
] | [
"Definition:Mapping",
"Definition:Restriction/Mapping",
"Definition:Image (Set Theory)/Mapping/Mapping",
"Definition:Cartesian Product"
] | [
"Restriction of Mapping is Mapping",
"Definition:Mapping"
] |
proofwiki-16269 | Primitive of Reciprocal of One plus Fourth Power of x | :$\ds \int \frac 1 {1 + x^4} \rd x = \frac 1 {2 \sqrt 2} \paren {\map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + \frac 1 2 \ln \size {\frac {x^2 + \sqrt 2 x + 1} {x^2 - \sqrt 2 x + 1} } } + C$ | From Primitive of $\dfrac {1 + x^2} {1 + x^4}$, we have:
:$\ds \int \frac {x^2 + 1} {x^4 + 1} \rd x = \frac 1 {\sqrt 2} \map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + C$
From Primitive of $\dfrac {-1 + x^2} {1 + x^4}$, we have:
:$\ds \int \frac {x^2 - 1} {x^4 + 1} \rd x = \frac 1 {2 \sqrt 2} \ln \size {\fra... | :$\ds \int \frac 1 {1 + x^4} \rd x = \frac 1 {2 \sqrt 2} \paren {\map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + \frac 1 2 \ln \size {\frac {x^2 + \sqrt 2 x + 1} {x^2 - \sqrt 2 x + 1} } } + C$ | From [[Primitive of One plus x Squared over One plus Fourth Power of x|Primitive of $\dfrac {1 + x^2} {1 + x^4}$]], we have:
:$\ds \int \frac {x^2 + 1} {x^4 + 1} \rd x = \frac 1 {\sqrt 2} \map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + C$
From [[Primitive of Minus One plus x Squared over One plus Fourth Po... | Primitive of Reciprocal of One plus Fourth Power of x/Proof 1 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_One_plus_Fourth_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_One_plus_Fourth_Power_of_x/Proof_1 | [
"Primitives involving Reciprocals",
"Primitives involving x to the fourth plus or minus a to the fourth",
"Primitive of Reciprocal of One plus Fourth Power of x"
] | [] | [
"Primitive of One plus x Squared over One plus Fourth Power of x",
"Primitive of Minus One plus x Squared over One plus Fourth Power of x",
"Logarithm of Reciprocal"
] |
proofwiki-16270 | Primitive of Reciprocal of One plus Fourth Power of x | :$\ds \int \frac 1 {1 + x^4} \rd x = \frac 1 {2 \sqrt 2} \paren {\map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + \frac 1 2 \ln \size {\frac {x^2 + \sqrt 2 x + 1} {x^2 - \sqrt 2 x + 1} } } + C$ | A special case of Primitive of $\dfrac 1 {x^4 + a^4}$, setting $a = 1$.
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^4 + a^4}
| r = \frac 1 {4 a^3 \sqrt 2} \map \ln {\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} } - \frac 1 {2 a^3 \sqrt 2} \paren {\map \arctan {1 - \frac {x \sqrt 2} a} - \map \arcta... | :$\ds \int \frac 1 {1 + x^4} \rd x = \frac 1 {2 \sqrt 2} \paren {\map \arctan {\frac 1 {\sqrt 2} \paren {x - \frac 1 x} } + \frac 1 2 \ln \size {\frac {x^2 + \sqrt 2 x + 1} {x^2 - \sqrt 2 x + 1} } } + C$ | A special case of [[Primitive of Reciprocal of x fourth plus a fourth|Primitive of $\dfrac 1 {x^4 + a^4}$]], setting $a = 1$.
{{begin-eqn}}
{{eqn | l = \int \frac {\d x} {x^4 + a^4}
| r = \frac 1 {4 a^3 \sqrt 2} \map \ln {\frac {x^2 + a x \sqrt 2 + a^2} {x^2 - a x \sqrt 2 + a^2} } - \frac 1 {2 a^3 \sqrt 2} \pare... | Primitive of Reciprocal of One plus Fourth Power of x/Proof 2 | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_One_plus_Fourth_Power_of_x | https://proofwiki.org/wiki/Primitive_of_Reciprocal_of_One_plus_Fourth_Power_of_x/Proof_2 | [
"Primitives involving Reciprocals",
"Primitives involving x to the fourth plus or minus a to the fourth",
"Primitive of Reciprocal of One plus Fourth Power of x"
] | [] | [
"Primitive of Reciprocal of x fourth plus a fourth",
"Difference of Arctangents",
"Difference of Two Squares",
"Inverse Tangent is Odd Function",
"Sum of Arctangent and Arccotangent",
"Inverse Tangent is Odd Function"
] |
proofwiki-16271 | Empty Set from Principle of Non-Contradiction | The empty set can be characterised as:
:$\O := \set {x: x \in E \text { and } x \notin E}$
where $E$ is an arbitrary set. | {{AimForCont}} $x \in \O$ as defined here.
Thus we have:
:$x \in E$
and:
:$x \notin E$
This is a contradiction.
It follows by Proof by Contradiction that $x \notin \O$.
Hence, as $x$ was arbitrary, there can be no $x$ such that $x \in \O$.
Thus $\O$ is the empty set by definition.
{{qed}} | The [[Definition:Empty Set|empty set]] can be characterised as:
:$\O := \set {x: x \in E \text { and } x \notin E}$
where $E$ is an arbitrary [[Definition:Set|set]]. | {{AimForCont}} $x \in \O$ as defined here.
Thus we have:
:$x \in E$
and:
:$x \notin E$
This is a [[Definition:Contradiction|contradiction]].
It follows by [[Proof by Contradiction]] that $x \notin \O$.
Hence, as $x$ was arbitrary, there can be no $x$ such that $x \in \O$.
Thus $\O$ is the [[Definition:Empty Set|e... | Empty Set from Principle of Non-Contradiction | https://proofwiki.org/wiki/Empty_Set_from_Principle_of_Non-Contradiction | https://proofwiki.org/wiki/Empty_Set_from_Principle_of_Non-Contradiction | [
"Empty Set"
] | [
"Definition:Empty Set",
"Definition:Set"
] | [
"Definition:Contradiction",
"Proof by Contradiction",
"Definition:Empty Set"
] |
proofwiki-16272 | Power Series Expansion for Sine Integral Function | {{begin-eqn}}
{{eqn | l = \map \Si x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1} \times \paren {2 n + 1}!}
| c =
}}
{{eqn | r = \dfrac x {1 \cdot 1!} - \dfrac {x^3} {3 \cdot 3!} + \dfrac {x^5} {5 \cdot 5!} - \dfrac {x^7} {7 \cdot 7!} + \cdots
| c =
}}
{{en... | {{begin-eqn}}
{{eqn | l = \map \Si x
| r = \int_0^x \frac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = \int_0^x \frac {\paren {-1}^n} u \paren {\sum_{n \mathop = 0}^\infty \frac {u^{2 n + 1} } {\paren {2 n + 1}!} } \rd u
| c = Power Series Expansion for Sine Function
}}
{{eqn... | {{begin-eqn}}
{{eqn | l = \map \Si x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1} \times \paren {2 n + 1}!}
| c =
}}
{{eqn | r = \dfrac x {1 \cdot 1!} - \dfrac {x^3} {3 \cdot 3!} + \dfrac {x^5} {5 \cdot 5!} - \dfrac {x^7} {7 \cdot 7!} + \cdots
| c =
}}
{{en... | {{begin-eqn}}
{{eqn | l = \map \Si x
| r = \int_0^x \frac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = \int_0^x \frac {\paren {-1}^n} u \paren {\sum_{n \mathop = 0}^\infty \frac {u^{2 n + 1} } {\paren {2 n + 1}!} } \rd u
| c = [[Power Series Expansion for Sine Function]]
}}
{... | Power Series Expansion for Sine Integral Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Sine_Integral_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Sine_Integral_Function | [
"Sine Integral Function",
"Examples of Power Series"
] | [] | [
"Power Series Expansion for Sine Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Primitive of Power"
] |
proofwiki-16273 | Sine Integral Function is Odd | :$\map \Si {-x} = -\map \Si x$ | {{begin-eqn}}
{{eqn | l = \map \Si {-x}
| r = \int_0^{-x} \frac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = -\int_0^x \frac {\map \sin {-u} } {-u} \rd u
| c = substituting $u \mapsto -u$
}}
{{eqn | r = -\int_0^x \frac {\sin u} u \rd u
| c = Sine Function is Odd
}}
{{eq... | :$\map \Si {-x} = -\map \Si x$ | {{begin-eqn}}
{{eqn | l = \map \Si {-x}
| r = \int_0^{-x} \frac {\sin u} u \rd u
| c = {{Defof|Sine Integral Function}}
}}
{{eqn | r = -\int_0^x \frac {\map \sin {-u} } {-u} \rd u
| c = [[Integration by Substitution|substituting]] $u \mapsto -u$
}}
{{eqn | r = -\int_0^x \frac {\sin u} u \rd u
| ... | Sine Integral Function is Odd | https://proofwiki.org/wiki/Sine_Integral_Function_is_Odd | https://proofwiki.org/wiki/Sine_Integral_Function_is_Odd | [
"Sine Integral Function",
"Examples of Odd Functions"
] | [] | [
"Integration by Substitution",
"Sine Function is Odd"
] |
proofwiki-16274 | Sine Integral Function of Zero | :$\map \Si 0 = 0$ | By Sine Integral Function is Odd, $\Si$ is an odd function.
Therefore, by Odd Function of Zero is Zero:
:$\map \Si 0 = 0$
{{qed}} | :$\map \Si 0 = 0$ | By [[Sine Integral Function is Odd]], $\Si$ is an [[Definition:Odd Function|odd function]].
Therefore, by [[Odd Function of Zero is Zero]]:
:$\map \Si 0 = 0$
{{qed}} | Sine Integral Function of Zero | https://proofwiki.org/wiki/Sine_Integral_Function_of_Zero | https://proofwiki.org/wiki/Sine_Integral_Function_of_Zero | [
"Sine Integral Function"
] | [] | [
"Sine Integral Function is Odd",
"Definition:Odd Function",
"Odd Function of Zero is Zero"
] |
proofwiki-16275 | Arctangent of Root 3 over 3 | :$\map \arctan {\dfrac {\sqrt 3} 3} = \dfrac \pi 6$ | By definition, $\arctan$ is the inverse of the tangent function's restriction to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
By {{tan|30}}:
:$\tan \dfrac \pi 6 = \dfrac {\sqrt 3} 3$
As $\dfrac \pi 6 \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$, we have by the definition of an inverse function:
:$\map \arctan {\dfrac {... | :$\map \arctan {\dfrac {\sqrt 3} 3} = \dfrac \pi 6$ | By [[Definition:Real Arctangent|definition]], $\arctan$ is the [[Definition:Inverse of Mapping|inverse]] of the [[Definition:Real Tangent Function|tangent function]]'s [[Definition:Restriction of Mapping|restriction]] to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
By {{tan|30}}:
:$\tan \dfrac \pi 6 = \dfrac {\sqrt 3}... | Arctangent of Root 3 over 3 | https://proofwiki.org/wiki/Arctangent_of_Root_3_over_3 | https://proofwiki.org/wiki/Arctangent_of_Root_3_over_3 | [
"Arctangent Function"
] | [] | [
"Definition:Inverse Tangent/Real/Arctangent",
"Definition:Inverse of Mapping",
"Definition:Tangent Function/Real",
"Definition:Restriction/Mapping",
"Definition:Inverse of Mapping",
"Category:Arctangent Function"
] |
proofwiki-16276 | Cardinality of Set Union/2 Sets | Let $S_1$ and $S_2$ be finite sets.
Then:
:$\card {S_1 \cup S_2} = \card {S_1} + \card {S_2} - \card {S_1 \cap S_2}$ | We have that Cardinality is Additive Function.
From Additive Function is Strongly Additive:
:$\card {S_1 \cup S_2} + \card {S_1 \cap S_2} = \card {S_1} + \card {S_2}$
from which the result follows.
{{qed}} | Let $S_1$ and $S_2$ be [[Definition:Finite Set|finite sets]].
Then:
:$\card {S_1 \cup S_2} = \card {S_1} + \card {S_2} - \card {S_1 \cap S_2}$ | We have that [[Cardinality is Additive Function]].
From [[Additive Function is Strongly Additive]]:
:$\card {S_1 \cup S_2} + \card {S_1 \cap S_2} = \card {S_1} + \card {S_2}$
from which the result follows.
{{qed}} | Cardinality of Set Union/2 Sets | https://proofwiki.org/wiki/Cardinality_of_Set_Union/2_Sets | https://proofwiki.org/wiki/Cardinality_of_Set_Union/2_Sets | [
"Cardinality of Set Union"
] | [
"Definition:Finite Set"
] | [
"Cardinality is Additive Function",
"Additive Function is Strongly Additive"
] |
proofwiki-16277 | Cardinality of Set Union/3 Sets | Let $S_1$, $S_2$ and $S_3$ be finite sets.
Then:
{{begin-eqn}}
{{eqn | l = \card {S_1 \cup S_2 \cup S_3}
| r = \card {S_1} + \card {S_2} + \card {S_3}
| c =
}}
{{eqn | o =
| ro= -
| r = \card {S_1 \cap S_2} - \card {S_1 \cap S_3} - \card {S_2 \cap S_3}
| c =
}}
{{eqn | o =
| r... | This is a specific example of Cardinality of Set Union: General Case.
{{qed}} | Let $S_1$, $S_2$ and $S_3$ be [[Definition:Finite Set|finite sets]].
Then:
{{begin-eqn}}
{{eqn | l = \card {S_1 \cup S_2 \cup S_3}
| r = \card {S_1} + \card {S_2} + \card {S_3}
| c =
}}
{{eqn | o =
| ro= -
| r = \card {S_1 \cap S_2} - \card {S_1 \cap S_3} - \card {S_2 \cap S_3}
| c... | This is a specific example of [[Cardinality of Set Union/General Case|Cardinality of Set Union: General Case]].
{{qed}} | Cardinality of Set Union/3 Sets | https://proofwiki.org/wiki/Cardinality_of_Set_Union/3_Sets | https://proofwiki.org/wiki/Cardinality_of_Set_Union/3_Sets | [
"Cardinality of Set Union"
] | [
"Definition:Finite Set"
] | [
"Cardinality of Set Union/General Case"
] |
proofwiki-16278 | Definite Integral to Infinity of Exponential of -i x^2 | :$\ds \int_0^\infty \map \exp {-i x^2} \rd x = \frac 1 2 \sqrt {\frac \pi 2} \paren {1 - i}$ | Let $R$ be a positive real number.
Let $C_1$ be the straight line segment from $0$ to $R$.
Let $C_2$ be the arc of the circle of radius $R$ centred at the origin connecting $R$ and $R e^{i \pi/4}$ anticlockwise.
Let $C_3$ be the straight line segment from $R e^{i \pi/4}$ to $0$.
{{explain|What is the context of $C_1$... | :$\ds \int_0^\infty \map \exp {-i x^2} \rd x = \frac 1 2 \sqrt {\frac \pi 2} \paren {1 - i}$ | Let $R$ be a [[Definition:Positive Real Number|positive real number]].
Let $C_1$ be the [[Definition:Straight Line|straight line]] segment from $0$ to $R$.
Let $C_2$ be the [[Definition:Arc of Circle|arc of the circle]] of radius $R$ centred at the origin connecting $R$ and $R e^{i \pi/4}$ [[Definition:Anticlockwis... | Definite Integral to Infinity of Exponential of -i x^2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-i_x^2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Exponential_of_-i_x^2 | [
"Definite Integrals involving Exponential Function"
] | [] | [
"Definition:Positive/Real Number",
"Definition:Line/Straight Line",
"Definition:Circle/Arc",
"Definition:Anticlockwise",
"Definition:Line/Straight Line",
"Complex Exponential Function is Entire",
"Definition:Holomorphic Function",
"Definition:Interior (Topology)",
"Definition:Region/Plane",
"Cauch... |
proofwiki-16279 | P-adic Number times Integer Power of p is P-adic Integer | Let $p$ be a prime number.
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers.
Let $\Z_p$ be the $p$-adic integers.
Then:
:$\forall a \in \Q_p: \exists n \in \N: p^n a \in \Z_p$ | Let $a \in \Q_p$. | Let $p$ be a [[Definition:Prime Number|prime number]].
Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the [[Definition:Valued Field of P-adic Numbers|$p$-adic numbers]].
Let $\Z_p$ be the [[Definition:P-adic Integer|$p$-adic integers]].
Then:
:$\forall a \in \Q_p: \exists n \in \N: p^n a \in \Z_p$ | Let $a \in \Q_p$. | P-adic Number times Integer Power of p is P-adic Integer | https://proofwiki.org/wiki/P-adic_Number_times_Integer_Power_of_p_is_P-adic_Integer | https://proofwiki.org/wiki/P-adic_Number_times_Integer_Power_of_p_is_P-adic_Integer | [
"P-adic Number Theory"
] | [
"Definition:Prime Number",
"Definition:Valued Field of P-adic Numbers",
"Definition:P-adic Integer"
] | [] |
proofwiki-16280 | Definite Integral to Infinity of Sine of a x^2 | :$\ds \int_0^\infty \map \sin {a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi {2 a} }$ | We have, by Euler's Formula: Corollary:
:$\map \exp {-i a x^2} = -i \map \sin {a x^2} + \map \cos {a x^2}$
As $\map \sin {a x^2}$ and $\map \cos {a x^2}$ are both real for real $a, x$, we therefore have:
{{begin-eqn}}
{{eqn | l = \int_0^\infty \map \sin {a x^2} \rd x
| r = -\int_0^\infty \map \Im {\map \exp {-i a x^2}... | :$\ds \int_0^\infty \map \sin {a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi {2 a} }$ | We have, by [[Euler's Formula/Corollary|Euler's Formula: Corollary]]:
:$\map \exp {-i a x^2} = -i \map \sin {a x^2} + \map \cos {a x^2}$
As $\map \sin {a x^2}$ and $\map \cos {a x^2}$ are both [[Definition:Real Number|real]] for real $a, x$, we therefore have:
{{begin-eqn}}
{{eqn | l = \int_0^\infty \map \sin {a x^2... | Definite Integral to Infinity of Sine of a x^2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Sine_of_a_x^2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Sine_of_a_x^2 | [
"Definite Integrals involving Sine Function"
] | [] | [
"Euler's Formula/Corollary",
"Definition:Real Number",
"Integration by Substitution",
"Definite Integral to Infinity of Exponential of -i x^2"
] |
proofwiki-16281 | Definite Integral to Infinity of Cosine of a x^2 | :$\ds \int_0^\infty \map \cos {a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi {2 a} }$ | We have, by Euler's Formula: Corollary:
:$\map \exp {-i a x^2} = -i \map \sin {a x^2} + \map \cos {a x^2}$
As $\map \sin {a x^2}$ and $\map \cos {a x^2}$ are both real for real $a, x$, we therefore have:
{{begin-eqn}}
{{eqn | l = \int_0^\infty \map \cos {a x^2} \rd x
| r = \int_0^\infty \map \Re {\map \exp {-i a x^2} ... | :$\ds \int_0^\infty \map \cos {a x^2} \rd x = \frac 1 2 \sqrt {\frac \pi {2 a} }$ | We have, by [[Euler's Formula/Corollary|Euler's Formula: Corollary]]:
:$\map \exp {-i a x^2} = -i \map \sin {a x^2} + \map \cos {a x^2}$
As $\map \sin {a x^2}$ and $\map \cos {a x^2}$ are both [[Definition:Real Number|real]] for real $a, x$, we therefore have:
{{begin-eqn}}
{{eqn | l = \int_0^\infty \map \cos {a x^2... | Definite Integral to Infinity of Cosine of a x^2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cosine_of_a_x^2 | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cosine_of_a_x^2 | [
"Definite Integrals involving Sine Function"
] | [] | [
"Euler's Formula/Corollary",
"Definition:Real Number",
"Integration by Substitution",
"Definite Integral to Infinity of Exponential of -i x^2"
] |
proofwiki-16282 | Definite Integral from 0 to 2 Pi of Reciprocal of One minus 2 a Cosine x plus a Squared | :$\ds \int_0^{2 \pi} \frac {\d x} {1 - 2 a \cos x + a^2} = \frac {2 \pi} {1 - a^2}$ | {{explain|The context of this needs to be explained a little more deeply: the integrand is defined as a real function, but the analysis is actually in the complex plane. The latter needs to be brought forward so as to make it clear to the reader.}}
Let $C$ be the unit open disk centred at $0$.
The boundary of $C$, $\p... | :$\ds \int_0^{2 \pi} \frac {\d x} {1 - 2 a \cos x + a^2} = \frac {2 \pi} {1 - a^2}$ | {{explain|The context of this needs to be explained a little more deeply: the integrand is defined as a real function, but the analysis is actually in the complex plane. The latter needs to be brought forward so as to make it clear to the reader.}}
Let $C$ be the unit [[Definition:Open Complex Disk|open disk]] centred... | Definite Integral from 0 to 2 Pi of Reciprocal of One minus 2 a Cosine x plus a Squared | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Reciprocal_of_One_minus_2_a_Cosine_x_plus_a_Squared | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Reciprocal_of_One_minus_2_a_Cosine_x_plus_a_Squared | [
"Definite Integrals involving Cosine Function"
] | [] | [
"Definition:Complex Disk/Open",
"Definition:Boundary",
"Definition:Parameterization",
"Euler's Cosine Identity",
"Derivative of Exponential Function",
"Definition:Integration/Integrand",
"Definition:Isolated Singularity/Pole",
"Definition:Complex Disk/Closed",
"Definition:Isolated Singularity/Pole",... |
proofwiki-16283 | Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x minus One | :$\ds \int_0^\infty \frac {\sinh a x} {e^{b x} - 1} \rd x = \frac 1 {2 a} - \frac \pi {2 b} \cot \frac {a \pi} b$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sinh a x} {e^{b x} - 1} \rd x
| r = \frac 1 2 \int_0^\infty \frac {e^{-b x} \paren {e^{a x} - e^{-a x} } } {1 - e^{-b x} }
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \int_0^\infty \paren {e^{\paren {a - b} x} - e^{-\paren {a + b} x} } \sum_{n \mathop = 0}^... | :$\ds \int_0^\infty \frac {\sinh a x} {e^{b x} - 1} \rd x = \frac 1 {2 a} - \frac \pi {2 b} \cot \frac {a \pi} b$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sinh a x} {e^{b x} - 1} \rd x
| r = \frac 1 2 \int_0^\infty \frac {e^{-b x} \paren {e^{a x} - e^{-a x} } } {1 - e^{-b x} }
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \int_0^\infty \paren {e^{\paren {a - b} x} - e^{-\paren {a + b} x} } \sum_{n \mathop = 0}^... | Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x minus One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Hyperbolic_Sine_of_a_x_over_Exponential_of_b_x_minus_One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Hyperbolic_Sine_of_a_x_over_Exponential_of_b_x_minus_One | [
"Definite Integrals involving Hyperbolic Sine Function"
] | [] | [
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Derivative of Exponential Function/Corollary 1",
"Definition:Positive/Integer",
"Exponential Tends to Zero and Infinity",
"Exponential of Zero",
"Difference of Two Squares",
"Mittag-Leffler Expansion for Cotangent Function",
"Definition:Inte... |
proofwiki-16284 | Intersection is Empty and Union is Universe if Sets are Complementary | Let $A$ and $B$ be subsets of a universe $\Bbb U$.
Then:
:$A \cap B = \O$ and $A \cup B = \Bbb U$
{{iff}}:
:$B = \relcomp {\Bbb U} A$
where $\relcomp {\Bbb U} A$ denotes the complement of $A$ with respect to $\Bbb U$. | From Complement Union with Superset is Universe: Corollary:
:$A \cup B = \mathbb U \iff \relcomp {\Bbb U} A \subseteq B$
and from Empty Intersection iff Subset of Complement:
:$A \cap B = \O \iff B \subseteq \relcomp {\Bbb U} A$
The result follows by definition of set equality.
{{qed}} | Let $A$ and $B$ be [[Definition:Subset|subsets]] of a [[Definition:Universal Set|universe]] $\Bbb U$.
Then:
:$A \cap B = \O$ and $A \cup B = \Bbb U$
{{iff}}:
:$B = \relcomp {\Bbb U} A$
where $\relcomp {\Bbb U} A$ denotes the [[Definition:Set Complement|complement]] of $A$ with respect to $\Bbb U$. | From [[Complement Union with Superset is Universe/Corollary|Complement Union with Superset is Universe: Corollary]]:
:$A \cup B = \mathbb U \iff \relcomp {\Bbb U} A \subseteq B$
and from [[Empty Intersection iff Subset of Complement]]:
:$A \cap B = \O \iff B \subseteq \relcomp {\Bbb U} A$
The result follows by defi... | Intersection is Empty and Union is Universe if Sets are Complementary | https://proofwiki.org/wiki/Intersection_is_Empty_and_Union_is_Universe_if_Sets_are_Complementary | https://proofwiki.org/wiki/Intersection_is_Empty_and_Union_is_Universe_if_Sets_are_Complementary | [
"Set Intersection",
"Set Union",
"Set Complement",
"Empty Set",
"Universal Set"
] | [
"Definition:Subset",
"Definition:Universal Set",
"Definition:Set Complement"
] | [
"Complement Union with Superset is Universe/Corollary",
"Empty Intersection iff Subset of Complement",
"Definition:Set Equality"
] |
proofwiki-16285 | Mittag-Leffler Expansion for Square of Secant Function | :$\ds \pi^2 \map {\sec^2} {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac {\paren {2 n + 1} + 4 z^2} {\paren {\paren {2 n + 1}^2 - 4 z^2}^2}$ | {{ProofWanted}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}} | :$\ds \pi^2 \map {\sec^2} {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac {\paren {2 n + 1} + 4 z^2} {\paren {\paren {2 n + 1}^2 - 4 z^2}^2}$ | {{ProofWanted}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}} | Mittag-Leffler Expansion for Square of Secant Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Square_of_Secant_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Square_of_Secant_Function | [
"Mittag-Leffler Expansions",
"Secant Function"
] | [] | [] |
proofwiki-16286 | Union with Intersection equals Intersection with Union iff Subset | Let $A$, $B$ and $C$ be sets.
Then:
:$\paren {A \cap B} \cup C = A \cap \paren {B \cup C} \iff C \subseteq A$ | === Necessary Condition ===
Let $C \subseteq A$.
We have:
{{begin-eqn}}
{{eqn | l = C
| o = \subseteq
| r = A
| c =
}}
{{eqn | ll= \leadsto
| l = A \cap C
| r = C
| c = Intersection with Subset is Subset
}}
{{eqn | ll= \leadsto
| l = \paren {A \cap B} \cup \paren {A \cap C}
... | Let $A$, $B$ and $C$ be [[Definition:Set|sets]].
Then:
:$\paren {A \cap B} \cup C = A \cap \paren {B \cup C} \iff C \subseteq A$ | === Necessary Condition ===
Let $C \subseteq A$.
We have:
{{begin-eqn}}
{{eqn | l = C
| o = \subseteq
| r = A
| c =
}}
{{eqn | ll= \leadsto
| l = A \cap C
| r = C
| c = [[Intersection with Subset is Subset]]
}}
{{eqn | ll= \leadsto
| l = \paren {A \cap B} \cup \paren {A \ca... | Union with Intersection equals Intersection with Union iff Subset | https://proofwiki.org/wiki/Union_with_Intersection_equals_Intersection_with_Union_iff_Subset | https://proofwiki.org/wiki/Union_with_Intersection_equals_Intersection_with_Union_iff_Subset | [
"Set Union",
"Set Intersection",
"Subsets"
] | [
"Definition:Set"
] | [
"Intersection with Subset is Subset",
"Intersection Distributes over Union",
"Union with Superset is Superset",
"Intersection Distributes over Union"
] |
proofwiki-16287 | Mittag-Leffler Expansion for Square of Cosecant Function | :$\ds \pi^2 \map {\csc^2} {\pi z} = \frac 1 {z^2} + 2 \sum_{n \mathop = 1}^\infty \frac {z^2 + n^2} {\paren {z^2 - n^2}^2}$ | {{ProofWanted}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}} | :$\ds \pi^2 \map {\csc^2} {\pi z} = \frac 1 {z^2} + 2 \sum_{n \mathop = 1}^\infty \frac {z^2 + n^2} {\paren {z^2 - n^2}^2}$ | {{ProofWanted}}
{{Namedfor|Magnus Gustaf Mittag-Leffler|cat = Mittag-Leffler}} | Mittag-Leffler Expansion for Square of Cosecant Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Square_of_Cosecant_Function | https://proofwiki.org/wiki/Mittag-Leffler_Expansion_for_Square_of_Cosecant_Function | [
"Mittag-Leffler Expansions",
"Cosecant Function"
] | [] | [] |
proofwiki-16288 | (A cap C) cup (B cap Complement C) = Empty iff B subset C subset Complement A | Let $A$, $B$ and $C$ be subsets of a universe $\Bbb U$.
Then:
:$\paren {A \cap C} \cup \paren {B \cap \map \complement C} = \O \iff B \subseteq C \subseteq \map \complement A$
where $\map \complement C$ denotes the complement of $C$ in $\Bbb U$. | {{begin-eqn}}
{{eqn | l = \paren {A \cap C} \cup \paren {B \cap \map \complement C}
| r = \O
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = A \cap C
| r = \O
| c = Union is Empty iff Sets are Empty
}}
{{eqn | lo= \land
| l = B \cap \map \complement C
| r = \O
| c =
}}
{{eqn... | Let $A$, $B$ and $C$ be [[Definition:Subset|subsets]] of a [[Definition:Universal Set|universe]] $\Bbb U$.
Then:
:$\paren {A \cap C} \cup \paren {B \cap \map \complement C} = \O \iff B \subseteq C \subseteq \map \complement A$
where $\map \complement C$ denotes the [[Definition:Set Complement|complement]] of $C$ in $\... | {{begin-eqn}}
{{eqn | l = \paren {A \cap C} \cup \paren {B \cap \map \complement C}
| r = \O
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = A \cap C
| r = \O
| c = [[Union is Empty iff Sets are Empty]]
}}
{{eqn | lo= \land
| l = B \cap \map \complement C
| r = \O
| c =
}}
{... | (A cap C) cup (B cap Complement C) = Empty iff B subset C subset Complement A | https://proofwiki.org/wiki/(A_cap_C)_cup_(B_cap_Complement_C)_=_Empty_iff_B_subset_C_subset_Complement_A | https://proofwiki.org/wiki/(A_cap_C)_cup_(B_cap_Complement_C)_=_Empty_iff_B_subset_C_subset_Complement_A | [
"Set Union",
"Set Intersection",
"Set Complement",
"Empty Set",
"Subsets"
] | [
"Definition:Subset",
"Definition:Universal Set",
"Definition:Set Complement"
] | [
"Union is Empty iff Sets are Empty",
"Empty Intersection iff Subset of Complement",
"Intersection with Complement is Empty iff Subset"
] |
proofwiki-16289 | Union of Intersections of 2 from 3 equals Intersection of Unions of 2 from 3 | Let $A$, $B$ and $C$ be sets.
Then:
:$\paren {A \cap B} \cup \paren {B \cap C} \cup \paren {C \cap A} = \paren {A \cup B} \cap \paren {B \cup C} \cap \paren {C \cup A}$ | {{begin-eqn}}
{{eqn | l = \paren {A \cap B} \cup \paren {B \cap C} \cup \paren {C \cap A}
| r = \paren {B \cap \paren {A \cup C} } \cup \paren {C \cap A}
| c = Intersection Distributes over Union
}}
{{eqn | r = \paren {B \cup \paren {C \cap A} } \cap \paren {\paren {A \cup C} \cup \paren {C \cap A} }
... | Let $A$, $B$ and $C$ be [[Definition:Set|sets]].
Then:
:$\paren {A \cap B} \cup \paren {B \cap C} \cup \paren {C \cap A} = \paren {A \cup B} \cap \paren {B \cup C} \cap \paren {C \cup A}$ | {{begin-eqn}}
{{eqn | l = \paren {A \cap B} \cup \paren {B \cap C} \cup \paren {C \cap A}
| r = \paren {B \cap \paren {A \cup C} } \cup \paren {C \cap A}
| c = [[Intersection Distributes over Union]]
}}
{{eqn | r = \paren {B \cup \paren {C \cap A} } \cap \paren {\paren {A \cup C} \cup \paren {C \cap A} }
... | Union of Intersections of 2 from 3 equals Intersection of Unions of 2 from 3 | https://proofwiki.org/wiki/Union_of_Intersections_of_2_from_3_equals_Intersection_of_Unions_of_2_from_3 | https://proofwiki.org/wiki/Union_of_Intersections_of_2_from_3_equals_Intersection_of_Unions_of_2_from_3 | [
"Set Intersection",
"Set Union"
] | [
"Definition:Set"
] | [
"Intersection Distributes over Union",
"Union Distributes over Intersection",
"Intersection is Subset of Union",
"Union with Superset is Superset",
"Union Distributes over Intersection"
] |
proofwiki-16290 | Definite Integral to Infinity of Sine of a x over Hyperbolic Sine of b x | :$\ds \int_0^\infty \frac {\sin a x} {\sinh b x} \rd x = \frac \pi {2 b} \tanh \frac {a \pi} {2 b}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sin a x} {\sinh b x} \rd x
| r = \frac 1 i \int_0^\infty \frac {e^{-b x} \paren {e^{i a x} - e^{-i a x} } } {1 - e^{-2 b x} } \rd x
| c = Euler's Sine Identity, {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 i \int_0^\infty \paren {e^{\paren {i a - b} x} - e... | :$\ds \int_0^\infty \frac {\sin a x} {\sinh b x} \rd x = \frac \pi {2 b} \tanh \frac {a \pi} {2 b}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sin a x} {\sinh b x} \rd x
| r = \frac 1 i \int_0^\infty \frac {e^{-b x} \paren {e^{i a x} - e^{-i a x} } } {1 - e^{-2 b x} } \rd x
| c = [[Euler's Sine Identity]], {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 i \int_0^\infty \paren {e^{\paren {i a - b} x}... | Definite Integral to Infinity of Sine of a x over Hyperbolic Sine of b x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Sine_of_a_x_over_Hyperbolic_Sine_of_b_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Sine_of_a_x_over_Hyperbolic_Sine_of_b_x | [
"Definite Integrals involving Hyperbolic Sine Function",
"Definite Integrals involving Sine Function"
] | [] | [
"Euler's Sine Identity",
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Primitive of Exponential of a x",
"Modulus of Limit",
"Exponential of Sum",
"Exponential Tends to Zero and Infinity",
"Modulus of Limit",
"Exponential of Sum",
"Exponential Tends to Zero and Infinity",
"Exponenti... |
proofwiki-16291 | Definite Integral to Infinity of Cosine of a x over Hyperbolic Cosine of b x | :$\ds \int_0^\infty \frac {\cos a x} {\cosh b x} \rd x = \frac \pi {2 b} \sech \frac {a \pi} {2 b}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\cos a x} {\cosh b x} \rd x
| r = \int_0^\infty \frac {e^{-b x} \paren {e^{i a x} + e^{-i a x} } } {1 - \paren {-e^{-2 b x} } } \rd x
| c = Euler's Cosine Identity, {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \int_0^\infty \paren {e^{\paren {i a - b} x} + e^{-\p... | :$\ds \int_0^\infty \frac {\cos a x} {\cosh b x} \rd x = \frac \pi {2 b} \sech \frac {a \pi} {2 b}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\cos a x} {\cosh b x} \rd x
| r = \int_0^\infty \frac {e^{-b x} \paren {e^{i a x} + e^{-i a x} } } {1 - \paren {-e^{-2 b x} } } \rd x
| c = [[Euler's Cosine Identity]], {{Defof|Hyperbolic Cosine}}
}}
{{eqn | r = \int_0^\infty \paren {e^{\paren {i a - b} x} + e^... | Definite Integral to Infinity of Cosine of a x over Hyperbolic Cosine of b x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cosine_of_a_x_over_Hyperbolic_Cosine_of_b_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Cosine_of_a_x_over_Hyperbolic_Cosine_of_b_x | [
"Definite Integrals involving Cosine Function",
"Definite Integrals involving Hyperbolic Cosine Function"
] | [] | [
"Euler's Cosine Identity",
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Primitive of Exponential of a x",
"Modulus of Limit",
"Exponential of Sum",
"Exponential Tends to Zero and Infinity",
"Modulus of Limit",
"Exponential of Sum",
"Exponential Tends to Zero and Infinity",
"Exponen... |
proofwiki-16292 | Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x plus One | :$\ds \int_0^\infty \frac {\sinh a x} {e^{b x} + 1} \rd x = \frac \pi {2 b} \csc \frac {a \pi} b - \frac 1 {2 a}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sinh a x} {e^{b x} + 1} \rd x
| r = \frac 1 2 \int_0^\infty \frac {e^{-b x} \paren {e^{a x} - e^{-a x} } } {1 - \paren {-e^{-b x} } } \rd x
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \int_0^\infty \paren {e^{\paren {a - b} x} - e^{-\paren {a + b} x} } \par... | :$\ds \int_0^\infty \frac {\sinh a x} {e^{b x} + 1} \rd x = \frac \pi {2 b} \csc \frac {a \pi} b - \frac 1 {2 a}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sinh a x} {e^{b x} + 1} \rd x
| r = \frac 1 2 \int_0^\infty \frac {e^{-b x} \paren {e^{a x} - e^{-a x} } } {1 - \paren {-e^{-b x} } } \rd x
| c = {{Defof|Hyperbolic Sine}}
}}
{{eqn | r = \frac 1 2 \int_0^\infty \paren {e^{\paren {a - b} x} - e^{-\paren {a + b} x} } \par... | Definite Integral to Infinity of Hyperbolic Sine of a x over Exponential of b x plus One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Hyperbolic_Sine_of_a_x_over_Exponential_of_b_x_plus_One | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Hyperbolic_Sine_of_a_x_over_Exponential_of_b_x_plus_One | [
"Definite Integrals involving Hyperbolic Sine Function",
"Definite Integrals involving Exponential Function"
] | [] | [
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Primitive of Exponential of a x",
"Definition:Positive/Integer",
"Exponential Tends to Zero and Infinity",
"Exponential of Zero",
"Difference of Two Squares",
"Mittag-Leffler Expansion for Cosecant Function"
] |
proofwiki-16293 | Definite Integral from 0 to 2 Pi of Reciprocal of Square of a plus b Sine x | :$\ds \int_0^{2 \pi} \frac {\d x} {\paren {a + b \sin x}^2} = \frac {2 \pi a} {\paren {a^2 - b^2}^{3/2} }$ | From Definite Integral from $0$ to $2 \pi$ of $\dfrac 1 {a + b \sin x}$, we have:
:$\ds \int_0^{2 \pi} \frac {\d x} {a + b \sin x} = \frac {2 \pi} {\sqrt {a^2 - b^2} }$
We have:
{{begin-eqn}}
{{eqn | l = \frac \partial {\partial a} \int_0^{2 \pi} \frac {\d x} {a + b \sin x}
| r = \int_0^{2 \pi} \frac \partial {\part... | :$\ds \int_0^{2 \pi} \frac {\d x} {\paren {a + b \sin x}^2} = \frac {2 \pi a} {\paren {a^2 - b^2}^{3/2} }$ | From [[Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Sine x|Definite Integral from $0$ to $2 \pi$ of $\dfrac 1 {a + b \sin x}$]], we have:
:$\ds \int_0^{2 \pi} \frac {\d x} {a + b \sin x} = \frac {2 \pi} {\sqrt {a^2 - b^2} }$
We have:
{{begin-eqn}}
{{eqn | l = \frac \partial {\partial a} \int_0^{2 \pi... | Definite Integral from 0 to 2 Pi of Reciprocal of Square of a plus b Sine x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Reciprocal_of_Square_of_a_plus_b_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_2_Pi_of_Reciprocal_of_Square_of_a_plus_b_Sine_x | [
"Definite Integrals involving Cosine Function"
] | [] | [
"Definite Integral from 0 to 2 Pi of Reciprocal of a plus b Sine x",
"Definite Integral of Partial Derivative",
"Quotient Rule for Derivatives",
"Quotient Rule for Derivatives"
] |
proofwiki-16294 | Definite Integral to Infinity of Reciprocal of Exponential of x minus One minus Exponential of -x over x | :$\ds \int_0^\infty \paren {\frac 1 {e^x - 1} - \frac {e^{-x} } x} \rd x = \gamma$ | {{begin-eqn}}
{{eqn | l = \gamma
| r = \lim_{n \mathop \to \infty} \paren {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}
| c = {{Defof|Euler-Mascheroni Constant}}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \paren {\sum_{k \mathop = 1}^n \paren {\int_0^1 x^{k - 1} \rd x } + \int_0^1 \frac {1 - x^{n - 1} } {\ln x... | :$\ds \int_0^\infty \paren {\frac 1 {e^x - 1} - \frac {e^{-x} } x} \rd x = \gamma$ | {{begin-eqn}}
{{eqn | l = \gamma
| r = \lim_{n \mathop \to \infty} \paren {\sum_{k \mathop = 1}^n \frac 1 k - \ln n}
| c = {{Defof|Euler-Mascheroni Constant}}
}}
{{eqn | r = \lim_{n \mathop \to \infty} \paren {\sum_{k \mathop = 1}^n \paren {\int_0^1 x^{k - 1} \rd x } + \int_0^1 \frac {1 - x^{n - 1} } {\ln x... | Definite Integral to Infinity of Reciprocal of Exponential of x minus One minus Exponential of -x over x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_Exponential_of_x_minus_One_minus_Exponential_of_-x_over_x | https://proofwiki.org/wiki/Definite_Integral_to_Infinity_of_Reciprocal_of_Exponential_of_x_minus_One_minus_Exponential_of_-x_over_x | [
"Definite Integrals involving Exponential Function",
"Euler-Mascheroni Constant"
] | [] | [
"Primitive of Power",
"Definite Integral from 0 to 1 of Difference of Powers of x over Logarithm of x",
"Linear Combination of Integrals/Definite",
"Sum of Infinite Geometric Sequence",
"Lebesgue's Dominated Convergence Theorem",
"Sequence of Powers of Number less than One",
"Derivative of Exponential F... |
proofwiki-16295 | Intersection Complement of Set with Itself is Complement | Let $A$ and $B$ be subsets of a universal set $\Bbb U$.
Let $\uparrow$ denote the operation on $A$ and $B$ defined as:
:$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$
where $\relcomp {\Bbb U} A$ denotes the complement of $A$ in $\Bbb U$.
Then:
:$A \uparrow A = \relcomp {\Bbb U} A$ | {{begin-eqn}}
{{eqn | l = A \uparrow A
| r = \relcomp {\Bbb U} {A \cap A}
| c = Definition of $\uparrow$
}}
{{eqn | r = \relcomp {\Bbb U} A
| c = Set Intersection is Idempotent
}}
{{end-eqn}}
{{qed}} | Let $A$ and $B$ be [[Definition:Subset|subsets]] of a [[Definition:Universal Set|universal set]] $\Bbb U$.
Let $\uparrow$ denote the [[Definition:Binary Operation|operation]] on $A$ and $B$ defined as:
:$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$
where $\relcomp {\Bbb U} A$ denotes the [[Defini... | {{begin-eqn}}
{{eqn | l = A \uparrow A
| r = \relcomp {\Bbb U} {A \cap A}
| c = Definition of $\uparrow$
}}
{{eqn | r = \relcomp {\Bbb U} A
| c = [[Set Intersection is Idempotent]]
}}
{{end-eqn}}
{{qed}} | Intersection Complement of Set with Itself is Complement | https://proofwiki.org/wiki/Intersection_Complement_of_Set_with_Itself_is_Complement | https://proofwiki.org/wiki/Intersection_Complement_of_Set_with_Itself_is_Complement | [
"Set Intersection",
"Set Complement"
] | [
"Definition:Subset",
"Definition:Universal Set",
"Definition:Operation/Binary Operation",
"Definition:Relative Complement"
] | [
"Set Intersection is Idempotent"
] |
proofwiki-16296 | Set Intersection expressed as Intersection Complement | Let $A$ and $B$ be subsets of a universal set $\Bbb U$.
Let $\uparrow$ denote the operation on $A$ and $B$ defined as:
:$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$
where $\relcomp {\Bbb U} A$ denotes the complement of $A$ in $\Bbb U$.
Then:
:$A \cap B = \paren {A \uparrow B} \uparrow \paren {A \... | {{begin-eqn}}
{{eqn | l = A \cap B
| r = \relcomp {\Bbb U} {\relcomp {\Bbb U} {A \cap B} }
| c = Relative Complement of Relative Complement
}}
{{eqn | r = \relcomp {\Bbb U} {A \uparrow B}
| c = Intersection Complement of Set with Itself is Complement
}}
{{eqn | r = \paren {A \uparrow B} \uparrow \pare... | Let $A$ and $B$ be [[Definition:Subset|subsets]] of a [[Definition:Universal Set|universal set]] $\Bbb U$.
Let $\uparrow$ denote the [[Definition:Binary Operation|operation]] on $A$ and $B$ defined as:
:$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$
where $\relcomp {\Bbb U} A$ denotes the [[Defini... | {{begin-eqn}}
{{eqn | l = A \cap B
| r = \relcomp {\Bbb U} {\relcomp {\Bbb U} {A \cap B} }
| c = [[Relative Complement of Relative Complement]]
}}
{{eqn | r = \relcomp {\Bbb U} {A \uparrow B}
| c = [[Intersection Complement of Set with Itself is Complement]]
}}
{{eqn | r = \paren {A \uparrow B} \uparr... | Set Intersection expressed as Intersection Complement | https://proofwiki.org/wiki/Set_Intersection_expressed_as_Intersection_Complement | https://proofwiki.org/wiki/Set_Intersection_expressed_as_Intersection_Complement | [
"Set Intersection",
"Set Complement"
] | [
"Definition:Subset",
"Definition:Universal Set",
"Definition:Operation/Binary Operation",
"Definition:Relative Complement"
] | [
"Relative Complement of Relative Complement",
"Intersection Complement of Set with Itself is Complement",
"Intersection Complement of Set with Itself is Complement"
] |
proofwiki-16297 | Set Union expressed as Intersection Complement | Let $A$ and $B$ be subsets of a universal set $\Bbb U$.
Let $\uparrow$ denote the operation on $A$ and $B$ defined as:
:$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$
where $\relcomp {\Bbb U} A$ denotes the complement of $A$ in $\Bbb U$.
Then:
:$A \cup B = \paren {A \uparrow A} \uparrow \paren {B \... | {{begin-eqn}}
{{eqn | l = A \cup B
| r = \relcomp {\Bbb U} {\relcomp {\Bbb U} A \cap \relcomp {\Bbb U} B}
| c = De Morgan's Laws : Complement of Union
}}
{{eqn | r = \relcomp {\Bbb U} {\paren {A \uparrow A} \cap \paren {B \uparrow B} }
| c = Intersection Complement of Set with Itself is Complement
}}
... | Let $A$ and $B$ be [[Definition:Subset|subsets]] of a [[Definition:Universal Set|universal set]] $\Bbb U$.
Let $\uparrow$ denote the [[Definition:Binary Operation|operation]] on $A$ and $B$ defined as:
:$\paren {A \uparrow B} \iff \paren {\relcomp {\Bbb U} {A \cap B} }$
where $\relcomp {\Bbb U} A$ denotes the [[Defini... | {{begin-eqn}}
{{eqn | l = A \cup B
| r = \relcomp {\Bbb U} {\relcomp {\Bbb U} A \cap \relcomp {\Bbb U} B}
| c = [[De Morgan's Laws (Set Theory)/Set Complement/Complement of Union|De Morgan's Laws : Complement of Union]]
}}
{{eqn | r = \relcomp {\Bbb U} {\paren {A \uparrow A} \cap \paren {B \uparrow B} }
... | Set Union expressed as Intersection Complement | https://proofwiki.org/wiki/Set_Union_expressed_as_Intersection_Complement | https://proofwiki.org/wiki/Set_Union_expressed_as_Intersection_Complement | [
"Set Union",
"Set Intersection",
"Set Complement"
] | [
"Definition:Subset",
"Definition:Universal Set",
"Definition:Operation/Binary Operation",
"Definition:Relative Complement"
] | [
"De Morgan's Laws (Set Theory)/Set Complement/Complement of Union",
"Intersection Complement of Set with Itself is Complement",
"Intersection Complement of Set with Itself is Complement"
] |
proofwiki-16298 | Element in Set iff Singleton in Powerset | Let $S$ be a set.
Then:
:$x \in S \iff \set x \in \powerset S$
where $\powerset S$ denotes the power set of $S$. | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = S
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \set x
| o = \subseteq
| r = S
| c = Singleton of Element is Subset
}}
{{eqn | ll= \leadstoandfrom
| l = \set x
| o = \in
| r = \powerset S
| c = {{Defof|Power Set}... | Let $S$ be a [[Definition:Set|set]].
Then:
:$x \in S \iff \set x \in \powerset S$
where $\powerset S$ denotes the [[Definition:Power Set|power set]] of $S$. | {{begin-eqn}}
{{eqn | l = x
| o = \in
| r = S
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = \set x
| o = \subseteq
| r = S
| c = [[Singleton of Element is Subset]]
}}
{{eqn | ll= \leadstoandfrom
| l = \set x
| o = \in
| r = \powerset S
| c = {{Defof|Power ... | Element in Set iff Singleton in Powerset | https://proofwiki.org/wiki/Element_in_Set_iff_Singleton_in_Powerset | https://proofwiki.org/wiki/Element_in_Set_iff_Singleton_in_Powerset | [
"Power Set",
"Singletons"
] | [
"Definition:Set",
"Definition:Power Set"
] | [
"Singleton of Element is Subset"
] |
proofwiki-16299 | Singleton of Subset is Element of Powerset of Powerset | Let $S \subseteq T$ where $S$ and $T$ are both sets.
Then:
:$\set S \in \powerset {\powerset T}$
where $\powerset T$ denotes the power set of $T$. | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T
| c =
}}
{{eqn | ll= \leadsto
| l = S
| o = \in
| r = \powerset T
| c = {{Defof|Power Set}}
}}
{{eqn | ll= \leadsto
| l = \set S
| o = \in
| r = \powerset {\powerset T}
| c = Element in Set iff Singleto... | Let $S \subseteq T$ where $S$ and $T$ are both [[Definition:Set|sets]].
Then:
:$\set S \in \powerset {\powerset T}$
where $\powerset T$ denotes the [[Definition:Power Set|power set]] of $T$. | {{begin-eqn}}
{{eqn | l = S
| o = \subseteq
| r = T
| c =
}}
{{eqn | ll= \leadsto
| l = S
| o = \in
| r = \powerset T
| c = {{Defof|Power Set}}
}}
{{eqn | ll= \leadsto
| l = \set S
| o = \in
| r = \powerset {\powerset T}
| c = [[Element in Set iff Single... | Singleton of Subset is Element of Powerset of Powerset | https://proofwiki.org/wiki/Singleton_of_Subset_is_Element_of_Powerset_of_Powerset | https://proofwiki.org/wiki/Singleton_of_Subset_is_Element_of_Powerset_of_Powerset | [
"Power Set",
"Subsets"
] | [
"Definition:Set",
"Definition:Power Set"
] | [
"Element in Set iff Singleton in Powerset"
] |
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