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-15900 | Laplace Transform of Cosine Integral Function | :$\laptrans {\map \Ci t} = \dfrac {\map \ln {s^2 + 1} } {2 s}$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\Ci$ denotes the cosine integral function. | Let $\map f t := \map \Ci t = \ds \int_t^\infty \dfrac {\cos u} u \rd u$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = -\dfrac {\cos t} t
| c =
}}
{{eqn | ll= \leadsto
| l = t \map {f'} t
| r = -\cos t
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {t \map {f'} t}
| r = -... | :$\laptrans {\map \Ci t} = \dfrac {\map \ln {s^2 + 1} } {2 s}$
where:
:$\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$
:$\Ci$ denotes the [[Definition:Cosine Integral Function|cosine integral function]]. | Let $\map f t := \map \Ci t = \ds \int_t^\infty \dfrac {\cos u} u \rd u$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = -\dfrac {\cos t} t
| c =
}}
{{eqn | ll= \leadsto
| l = t \map {f'} t
| r = -\cos t
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {t \map {f'} t}
| r = ... | Laplace Transform of Cosine Integral Function/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine_Integral_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Cosine_Integral_Function/Proof_1 | [
"Laplace Transform of Cosine Integral Function",
"Examples of Laplace Transforms",
"Cosine Integral Function"
] | [
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Cosine Integral Function"
] | [
"Laplace Transform of Cosine",
"Derivative of Laplace Transform",
"Laplace Transform of Derivative",
"Primitive of x over x squared plus a squared",
"Initial Value Theorem of Laplace Transform"
] |
proofwiki-15901 | Laplace Transform 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:
:$\laptrans {\map \Ei t} = \dfrac {\map \ln {s + 1} } s$
where $\laptrans f$ denotes the Laplace transform of the function $f$ | Let $\map f t := \map \Ei t = \ds \int_t^\infty \dfrac {e^{-u} } u \rd u$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = -\dfrac {e^{-t} } t
| c =
}}
{{eqn | ll= \leadsto
| l = t \map {f'} t
| r = -e^{-t}
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {t \map {f'} t}
| r =... | 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:
:$\laptrans {\map \Ei t} = \dfrac {\map \ln {s + 1} } s$
where $\laptrans f$ denotes the [[Defi... | Let $\map f t := \map \Ei t = \ds \int_t^\infty \dfrac {e^{-u} } u \rd u$.
Then:
{{begin-eqn}}
{{eqn | l = \map {f'} t
| r = -\dfrac {e^{-t} } t
| c =
}}
{{eqn | ll= \leadsto
| l = t \map {f'} t
| r = -e^{-t}
| c =
}}
{{eqn | ll= \leadsto
| l = \laptrans {t \map {f'} t}
| r ... | Laplace Transform of Exponential Integral Function/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential_Integral_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Exponential_Integral_Function/Proof_1 | [
"Laplace Transform of Exponential Integral Function",
"Exponential Integral Function",
"Examples of Laplace Transforms"
] | [
"Definition:Exponential Integral Function/Formulation 1",
"Definition:Laplace Transform",
"Definition:Real Function"
] | [
"Laplace Transform of Exponential",
"Derivative of Laplace Transform",
"Laplace Transform of Derivative",
"Primitive of Reciprocal of a x + b",
"Initial Value Theorem of Laplace Transform"
] |
proofwiki-15902 | Laplace Transform of Heaviside Step Function | Let $\map {u_c} t$ denote the Heaviside step function:
:<nowiki>$\map {u_c} t = \begin {cases}
1 & : t > c \\
0 & : t < c
\end {cases}$</nowiki>
The Laplace transform of $\map {u_c} t$ is given by:
:$\laptrans {\map {u_c} t} = \dfrac {e^{-s c} } s$
for $\map \Re s > c$. | {{begin-eqn}}
{{eqn | l = \laptrans {\map {u_c} t}
| r = \int_0^{\to +\infty} \map {u_c} t e^{-s t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^c \map {u_c} t e^{-s t} \rd t + \int_c^{\to +\infty} \map {u_c} t e^{-s t} \rd t
| c = Sum of Integrals on Adjacent Intervals for Integrable... | Let $\map {u_c} t$ denote the [[Definition:Heaviside Step Function|Heaviside step function]]:
:<nowiki>$\map {u_c} t = \begin {cases}
1 & : t > c \\
0 & : t < c
\end {cases}$</nowiki>
The [[Definition:Laplace Transform|Laplace transform]] of $\map {u_c} t$ is given by:
:$\laptrans {\map {u_c} t} = \dfrac {e^{-s c} ... | {{begin-eqn}}
{{eqn | l = \laptrans {\map {u_c} t}
| r = \int_0^{\to +\infty} \map {u_c} t e^{-s t} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^c \map {u_c} t e^{-s t} \rd t + \int_c^{\to +\infty} \map {u_c} t e^{-s t} \rd t
| c = [[Sum of Integrals on Adjacent Intervals for Integrab... | Laplace Transform of Heaviside Step Function/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Heaviside_Step_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Heaviside_Step_Function/Proof_1 | [
"Laplace Transform of Heaviside Step Function",
"Heaviside Step Function",
"Examples of Laplace Transforms"
] | [
"Definition:Heaviside Step Function",
"Definition:Laplace Transform"
] | [
"Sum of Integrals on Adjacent Intervals for Integrable Functions",
"Primitive of Exponential of a x"
] |
proofwiki-15903 | Laplace Transform of Heaviside Step Function | Let $\map {u_c} t$ denote the Heaviside step function:
:<nowiki>$\map {u_c} t = \begin {cases}
1 & : t > c \\
0 & : t < c
\end {cases}$</nowiki>
The Laplace transform of $\map {u_c} t$ is given by:
:$\laptrans {\map {u_c} t} = \dfrac {e^{-s c} } s$
for $\map \Re s > c$. | {{begin-eqn}}
{{eqn | l = \laptrans 1
| r = \dfrac 1 s
| c = Laplace Transform of 1
}}
{{eqn | ll= \leadsto
| l = \laptrans {1 \times \map {u_c} t}
| r = \dfrac 1 s \times e^{-c s}
| c = Second Translation Property of Laplace Transforms
}}
{{eqn | r = \dfrac {e^{-s c} } s
| c = simpl... | Let $\map {u_c} t$ denote the [[Definition:Heaviside Step Function|Heaviside step function]]:
:<nowiki>$\map {u_c} t = \begin {cases}
1 & : t > c \\
0 & : t < c
\end {cases}$</nowiki>
The [[Definition:Laplace Transform|Laplace transform]] of $\map {u_c} t$ is given by:
:$\laptrans {\map {u_c} t} = \dfrac {e^{-s c} ... | {{begin-eqn}}
{{eqn | l = \laptrans 1
| r = \dfrac 1 s
| c = [[Laplace Transform of 1]]
}}
{{eqn | ll= \leadsto
| l = \laptrans {1 \times \map {u_c} t}
| r = \dfrac 1 s \times e^{-c s}
| c = [[Second Translation Property of Laplace Transforms]]
}}
{{eqn | r = \dfrac {e^{-s c} } s
| c... | Laplace Transform of Heaviside Step Function/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Heaviside_Step_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Heaviside_Step_Function/Proof_2 | [
"Laplace Transform of Heaviside Step Function",
"Heaviside Step Function",
"Examples of Laplace Transforms"
] | [
"Definition:Heaviside Step Function",
"Definition:Laplace Transform"
] | [
"Laplace Transform of 1",
"Second Translation Property of Laplace Transforms"
] |
proofwiki-15904 | Laplace Transform of Dirac Delta Function | Let $\map \delta t$ denote the Dirac delta function.
The Laplace transform of $\map \delta t$ is given by:
:$\laptrans {\map \delta t} = 1$ | === Lemma ===
{{:Laplace Transform of Dirac Delta Function/Lemma}}{{qed|lemma}}
Then:
{{begin-eqn}}
{{eqn | l = \laptrans {\map \delta t}
| r = \lim_{\epsilon \mathop \to 0} \laptrans {\map {F_\epsilon} t}
| c = {{Defof|Dirac Delta Function|index = 1}}
}}
{{eqn | r = \lim_{\epsilon \mathop \to 0} \dfrac {1 ... | Let $\map \delta t$ denote the [[Definition:Dirac Delta Function|Dirac delta function]].
The [[Definition:Laplace Transform|Laplace transform]] of $\map \delta t$ is given by:
:$\laptrans {\map \delta t} = 1$ | === [[Laplace Transform of Dirac Delta Function/Lemma|Lemma]] ===
{{:Laplace Transform of Dirac Delta Function/Lemma}}{{qed|lemma}}
Then:
{{begin-eqn}}
{{eqn | l = \laptrans {\map \delta t}
| r = \lim_{\epsilon \mathop \to 0} \laptrans {\map {F_\epsilon} t}
| c = {{Defof|Dirac Delta Function|index = 1}}
}... | Laplace Transform of Dirac Delta Function/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Dirac_Delta_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Dirac_Delta_Function/Proof_2 | [
"Laplace Transform of Dirac Delta Function",
"Dirac Delta Function",
"Examples of Laplace Transforms"
] | [
"Definition:Dirac Delta Function",
"Definition:Laplace Transform"
] | [
"Laplace Transform of Dirac Delta Function/Lemma",
"Laplace Transform of Dirac Delta Function/Lemma"
] |
proofwiki-15905 | Laplace Transform of Dirac Delta Function | Let $\map \delta t$ denote the Dirac delta function.
The Laplace transform of $\map \delta t$ is given by:
:$\laptrans {\map \delta t} = 1$ | === Lemma ===
{{:Laplace Transform of Dirac Delta Function/Lemma}}{{qed|lemma}}
Then:
{{begin-eqn}}
{{eqn | l = \laptrans {\map \delta t}
| r = \lim_{\epsilon \mathop \to 0} \laptrans {\map {F_\epsilon} t}
| c = {{Defof|Dirac Delta Function|index = 1}}
}}
{{eqn | r = \lim_{\epsilon \mathop \to 0} \dfrac {1 ... | Let $\map \delta t$ denote the [[Definition:Dirac Delta Function|Dirac delta function]].
The [[Definition:Laplace Transform|Laplace transform]] of $\map \delta t$ is given by:
:$\laptrans {\map \delta t} = 1$ | === [[Laplace Transform of Dirac Delta Function/Lemma|Lemma]] ===
{{:Laplace Transform of Dirac Delta Function/Lemma}}{{qed|lemma}}
Then:
{{begin-eqn}}
{{eqn | l = \laptrans {\map \delta t}
| r = \lim_{\epsilon \mathop \to 0} \laptrans {\map {F_\epsilon} t}
| c = {{Defof|Dirac Delta Function|index = 1}}
}... | Laplace Transform of Dirac Delta Function/Proof 3 | https://proofwiki.org/wiki/Laplace_Transform_of_Dirac_Delta_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Dirac_Delta_Function/Proof_3 | [
"Laplace Transform of Dirac Delta Function",
"Dirac Delta Function",
"Examples of Laplace Transforms"
] | [
"Definition:Dirac Delta Function",
"Definition:Laplace Transform"
] | [
"Laplace Transform of Dirac Delta Function/Lemma",
"Laplace Transform of Dirac Delta Function/Lemma",
"L'Hôpital's Rule",
"Exponential of Zero"
] |
proofwiki-15906 | Laplace Transform of Shifted Dirac Delta Function | Let $\map \delta t$ denote the Dirac delta function.
The Laplace transform of $\map \delta {t - a}$ is given by:
:$\laptrans {\map \delta {t - a} } = e^{-a s}$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map \delta {t - a} }
| r = \int_0^{\to +\infty} e^{-s t} \map \delta {t - a} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = e^{-s \times a}
| c = Integral to Infinity of Shifted Dirac Delta Function by Continuous Function
}}
{{eqn | r = e^{-a s}
| c ... | Let $\map \delta t$ denote the [[Definition:Dirac Delta Function|Dirac delta function]].
The [[Definition:Laplace Transform|Laplace transform]] of $\map \delta {t - a}$ is given by:
:$\laptrans {\map \delta {t - a} } = e^{-a s}$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map \delta {t - a} }
| r = \int_0^{\to +\infty} e^{-s t} \map \delta {t - a} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = e^{-s \times a}
| c = [[Integral to Infinity of Shifted Dirac Delta Function by Continuous Function]]
}}
{{eqn | r = e^{-a s}
... | Laplace Transform of Shifted Dirac Delta Function/Proof 1 | https://proofwiki.org/wiki/Laplace_Transform_of_Shifted_Dirac_Delta_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Shifted_Dirac_Delta_Function/Proof_1 | [
"Laplace Transform of Shifted Dirac Delta Function",
"Dirac Delta Function",
"Examples of Laplace Transforms"
] | [
"Definition:Dirac Delta Function",
"Definition:Laplace Transform"
] | [
"Integral to Infinity of Shifted Dirac Delta Function by Continuous Function"
] |
proofwiki-15907 | Laplace Transform of Null Function | Let $\NN: \R \to \R$ be a null function.
The Laplace transform of $\map \NN t$ is given by:
:$\laptrans {\map \NN t} = 0$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map \NN t}
| r = \int_0^{\to +\infty} e^{-s t} \map \NN t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{end-eqn}}
{{finish}} | Let $\NN: \R \to \R$ be a [[Definition:Null Function|null function]].
The [[Definition:Laplace Transform|Laplace transform]] of $\map \NN t$ is given by:
:$\laptrans {\map \NN t} = 0$ | {{begin-eqn}}
{{eqn | l = \laptrans {\map \NN t}
| r = \int_0^{\to +\infty} e^{-s t} \map \NN t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{end-eqn}}
{{finish}} | Laplace Transform of Null Function | https://proofwiki.org/wiki/Laplace_Transform_of_Null_Function | https://proofwiki.org/wiki/Laplace_Transform_of_Null_Function | [
"Null Functions",
"Examples of Laplace Transforms"
] | [
"Definition:Null Function",
"Definition:Laplace Transform"
] | [] |
proofwiki-15908 | Second Translation Property of Laplace Transforms/Proof 1 | Let $f$ be a function such that $\laptrans f$ exists.
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.
Let $a \in \C$ or $\R$ be constant.
{{:Second Translation Property of Laplace Transforms}} | {{begin-eqn}}
{{eqn | l = \laptrans {\map f {t - a} }
| r = \int_0^{\to + \infty} e^{-s t} \map f {t - a} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to + \infty} e^{-s \paren {t - a} } e^{-a s} \map f {t - a} \rd \paren {t - a}
| c =
}}
{{eqn | r = e^{-a s} \int_0^{\to + \infty} ... | Let $f$ be a [[Definition:Function|function]] such that $\laptrans f$ exists.
Let $\laptrans {\map f t} = \map F s$ denote the [[Definition:Laplace Transform|Laplace transform]] of $f$.
Let $a \in \C$ or $\R$ be [[Definition:Constant|constant]].
{{:Second Translation Property of Laplace Transforms}} | {{begin-eqn}}
{{eqn | l = \laptrans {\map f {t - a} }
| r = \int_0^{\to + \infty} e^{-s t} \map f {t - a} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^{\to + \infty} e^{-s \paren {t - a} } e^{-a s} \map f {t - a} \rd \paren {t - a}
| c =
}}
{{eqn | r = e^{-a s} \int_0^{\to + \infty} ... | Second Translation Property of Laplace Transforms/Proof 1 | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms/Proof_1 | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms/Proof_1 | [
"Second Translation Property of Laplace Transforms"
] | [
"Definition:Function",
"Definition:Laplace Transform",
"Definition:Constant"
] | [] |
proofwiki-15909 | Second Translation Property of Laplace Transforms/Proof 2 | Let $f$ be a function such that $\laptrans f$ exists.
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of $f$.
Let $a \in \C$ or $\R$ be constant.
{{:Second Translation Property of Laplace Transforms}} | {{begin-eqn}}
{{eqn | l = \laptrans {\map g t}
| r = \int_0^\infty e^{-s t} \map g t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^a e^{-s t} \map g t \rd t + \int_a^\infty e^{-s t} \map g t \rd t
| c =
}}
{{eqn | r = \int_0^a 0 \times e^{-s t} \rd t + \int_a^\infty e^{-s t} \map f {t... | Let $f$ be a [[Definition:Function|function]] such that $\laptrans f$ exists.
Let $\laptrans {\map f t} = \map F s$ denote the [[Definition:Laplace Transform|Laplace transform]] of $f$.
Let $a \in \C$ or $\R$ be [[Definition:Constant|constant]].
{{:Second Translation Property of Laplace Transforms}} | {{begin-eqn}}
{{eqn | l = \laptrans {\map g t}
| r = \int_0^\infty e^{-s t} \map g t \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^a e^{-s t} \map g t \rd t + \int_a^\infty e^{-s t} \map g t \rd t
| c =
}}
{{eqn | r = \int_0^a 0 \times e^{-s t} \rd t + \int_a^\infty e^{-s t} \map f {t... | Second Translation Property of Laplace Transforms/Proof 2 | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms/Proof_2 | https://proofwiki.org/wiki/Second_Translation_Property_of_Laplace_Transforms/Proof_2 | [
"Second Translation Property of Laplace Transforms"
] | [
"Definition:Function",
"Definition:Laplace Transform",
"Definition:Constant"
] | [
"Integration by Substitution"
] |
proofwiki-15910 | Laplace Transform of Sine of t over t | Let $\sin$ denote the real sine function.
Let $\laptrans f$ denote the Laplace transform of a real function $f$.
Then:
:$\laptrans {\dfrac {\sin t} t} = \arctan \dfrac 1 s$ | From Limit of $\dfrac {\sin x} x$ at Zero:
:$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$
From Laplace Transform of Sine:
:$(1): \quad \laptrans {\sin t} = \dfrac 1 {s^2 + 1}$
From Laplace Transform of Integral:
:$(2): \quad \ds \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$
Hence:
{{begin-eq... | Let $\sin$ denote the [[Definition:Real Sine Function|real sine function]].
Let $\laptrans f$ denote the [[Definition:Laplace Transform|Laplace transform]] of a [[Definition:Real Function|real function]] $f$.
Then:
:$\laptrans {\dfrac {\sin t} t} = \arctan \dfrac 1 s$ | From [[Limit of Sine of X over X at Zero|Limit of $\dfrac {\sin x} x$ at Zero]]:
:$\ds \lim_{x \mathop \to 0} \frac {\sin x} x = 1$
From [[Laplace Transform of Sine]]:
:$(1): \quad \laptrans {\sin t} = \dfrac 1 {s^2 + 1}$
From [[Laplace Transform of Integral]]:
:$(2): \quad \ds \laptrans {\dfrac {\map f t} t} = \... | Laplace Transform of Sine of t over t | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_of_t_over_t | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_of_t_over_t | [
"Laplace Transform of Sine",
"Laplace Transforms involving Sine Function",
"Examples of Laplace Transforms"
] | [
"Definition:Sine/Real Function",
"Definition:Laplace Transform",
"Definition:Real Function"
] | [
"Limit of Sinc Function at Zero",
"Laplace Transform of Sine",
"Laplace Transform of Integral",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Sum of Arctangent and Arccotangent",
"Arctangent of Reciprocal equals Arccotangent"
] |
proofwiki-15911 | Laplace Transform of Sine of t over t/Corollary | :$\laptrans {\dfrac {\sin a t} t} = \arctan \dfrac a s$ | {{begin-eqn}}
{{eqn | l = \laptrans {\dfrac {\sin t} t}
| r = \arctan \dfrac 1 s
| c = Laplace Transform of Sine of t over t
}}
{{eqn | ll= \leadsto
| l = \laptrans {\dfrac {\sin a t} {a t} }
| r = \dfrac 1 a \arctan \dfrac 1 {s / a}
| c = Laplace Transform of Function of Constant Multiple... | :$\laptrans {\dfrac {\sin a t} t} = \arctan \dfrac a s$ | {{begin-eqn}}
{{eqn | l = \laptrans {\dfrac {\sin t} t}
| r = \arctan \dfrac 1 s
| c = [[Laplace Transform of Sine of t over t]]
}}
{{eqn | ll= \leadsto
| l = \laptrans {\dfrac {\sin a t} {a t} }
| r = \dfrac 1 a \arctan \dfrac 1 {s / a}
| c = [[Laplace Transform of Function of Constant Mu... | Laplace Transform of Sine of t over t/Corollary | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_of_t_over_t/Corollary | https://proofwiki.org/wiki/Laplace_Transform_of_Sine_of_t_over_t/Corollary | [
"Examples of Laplace Transforms",
"Sine Function",
"Laplace Transform of Sine"
] | [] | [
"Laplace Transform of Sine of t over t",
"Laplace Transform of Function of Constant Multiple"
] |
proofwiki-15912 | Integral to Infinity of Function over Argument | :$\ds \int_0^\infty {\dfrac {\map f t} t} = \int_0^{\to \infty} \map F u \rd u$
provided the integrals converge. | {{begin-eqn}}
{{eqn | l = \laptrans {\dfrac {\map f t} t}
| r = \int_s^{\to \infty} \map F u \rd u
| c = Integral of Laplace Transform
}}
{{eqn | ll= \leadsto
| l = \int_0^\infty e^{-s t} {\dfrac {\map f t} t} \rd t
| r = \int_s^{\to \infty} \map F u \rd u
| c = {{Defof|Laplace Transform}}... | :$\ds \int_0^\infty {\dfrac {\map f t} t} = \int_0^{\to \infty} \map F u \rd u$
provided the integrals [[Definition:Convergent Integral|converge]]. | {{begin-eqn}}
{{eqn | l = \laptrans {\dfrac {\map f t} t}
| r = \int_s^{\to \infty} \map F u \rd u
| c = [[Integral of Laplace Transform]]
}}
{{eqn | ll= \leadsto
| l = \int_0^\infty e^{-s t} {\dfrac {\map f t} t} \rd t
| r = \int_s^{\to \infty} \map F u \rd u
| c = {{Defof|Laplace Transfo... | Integral to Infinity of Function over Argument | https://proofwiki.org/wiki/Integral_to_Infinity_of_Function_over_Argument | https://proofwiki.org/wiki/Integral_to_Infinity_of_Function_over_Argument | [
"Laplace Transforms"
] | [
"Definition:Convergent Integral"
] | [
"Integral of Laplace Transform"
] |
proofwiki-15913 | Laplace Transform of Half Wave Rectified Sine Curve | Consider the half wave rectified sine curve:
:$\map f t = \begin {cases} \sin t & : 2 n \pi \le t \le \paren {2 n + 1} \pi \\ 0 & : \paren {2 n + 1} \pi \le t \le \paren {2 n + 2} \pi \end {cases}$
The Laplace transform of $\map f t$ is given by:
:$\laptrans {\map f t} = \dfrac 1 {\paren {1 - e^{-\pi s} } \paren {s^2 +... | We have that $\map f t$ is periodic with period $2 \pi$:
:800px
Hence:
{{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \dfrac 1 {1 - e^{-2 \pi s} } \int_0^{2 \pi} e^{-s t} \map f t \rd t
| c = Laplace Transform of Periodic Function
}}
{{eqn | r = \dfrac 1 {1 - e^{-2 \pi s} } \paren {\int_0^\pi e^{-s t}... | Consider the [[Definition:Half Wave Rectified Sine Curve|half wave rectified sine curve]]:
:$\map f t = \begin {cases} \sin t & : 2 n \pi \le t \le \paren {2 n + 1} \pi \\ 0 & : \paren {2 n + 1} \pi \le t \le \paren {2 n + 2} \pi \end {cases}$
The [[Definition:Laplace Transform|Laplace transform]] of $\map f t$ is g... | We have that $\map f t$ is [[Definition:Periodic Real Function|periodic]] with [[Definition:Period of Periodic Real Function|period]] $2 \pi$:
:[[File:Half-wave-rectified-sine-curve.png|800px]]
Hence:
{{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \dfrac 1 {1 - e^{-2 \pi s} } \int_0^{2 \pi} e^{-s t} \m... | Laplace Transform of Half Wave Rectified Sine Curve | https://proofwiki.org/wiki/Laplace_Transform_of_Half_Wave_Rectified_Sine_Curve | https://proofwiki.org/wiki/Laplace_Transform_of_Half_Wave_Rectified_Sine_Curve | [
"Half Wave Rectified Sine Curve",
"Examples of Laplace Transforms"
] | [
"Definition:Half Wave Rectified Sine Curve",
"Definition:Laplace Transform"
] | [
"Definition:Periodic Function/Real",
"Definition:Periodic Real Function/Period",
"File:Half-wave-rectified-sine-curve.png",
"Laplace Transform of Periodic Function",
"Primitive of Exponential of a x by Sine of b x",
"Sine of Integer Multiple of Pi",
"Cosine of Integer Multiple of Pi",
"Exponential of ... |
proofwiki-15914 | Integral to Infinity of Exponential of -t^2 | :$\ds \int_0^\infty \map \exp {-t^2} \rd t = \dfrac {\sqrt \pi} 2$ | Let $\ds I = \int_0^\infty \map \exp {-t^2} \rd t$.
Let $\ds I_P = \int_0^P \map \exp {-x^2} \rd x = \int_0^P \map \exp {-y^2} \rd y$.
Then we have:
:$I = \ds \lim_{P \mathop \to \infty} I_P$
Hence:
{{begin-eqn}}
{{eqn | l = {I_P}^2
| r = \paren {\int_0^P \map \exp {-x^2} \rd x} \paren {\int_0^P \map \exp {-y^2} ... | :$\ds \int_0^\infty \map \exp {-t^2} \rd t = \dfrac {\sqrt \pi} 2$ | Let $\ds I = \int_0^\infty \map \exp {-t^2} \rd t$.
Let $\ds I_P = \int_0^P \map \exp {-x^2} \rd x = \int_0^P \map \exp {-y^2} \rd y$.
Then we have:
:$I = \ds \lim_{P \mathop \to \infty} I_P$
Hence:
{{begin-eqn}}
{{eqn | l = {I_P}^2
| r = \paren {\int_0^P \map \exp {-x^2} \rd x} \paren {\int_0^P \map \exp {-y... | Integral to Infinity of Exponential of -t^2/Proof 1 | https://proofwiki.org/wiki/Integral_to_Infinity_of_Exponential_of_-t^2 | https://proofwiki.org/wiki/Integral_to_Infinity_of_Exponential_of_-t^2/Proof_1 | [
"Integral to Infinity of Exponential of -t^2",
"Gauss Error Function",
"Definite Integrals involving Exponential Function"
] | [] | [
"Definition:Quadrilateral/Square",
"File:Integral to Infinity of Exponential of -t^2.png",
"Definition:Integration/Integrand",
"Definition:Positive Real Function",
"Definition:Cartesian Plane/Quadrants/First",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:P... |
proofwiki-15915 | Integral to Infinity of Exponential of -t^2 | :$\ds \int_0^\infty \map \exp {-t^2} \rd t = \dfrac {\sqrt \pi} 2$ | Let $\lambda$ be a non-negative real number.
Then, we have:
:$\ds \size {\frac {e^{-\lambda^2 \paren {1 + x^2} } } {1 + x^2} } \le \frac 1 {1 + x^2}$
for each $x \in \R$.
Note that from Definite Integral to Infinity of $\dfrac 1 {x^2 + a^2}$:
:$\ds \int_0^\infty \frac 1 {x^2 + 1} \rd x = \frac \pi 2$
So by the Compa... | :$\ds \int_0^\infty \map \exp {-t^2} \rd t = \dfrac {\sqrt \pi} 2$ | Let $\lambda$ be a [[Definition:Non-Negative Real Number|non-negative real number]].
Then, we have:
:$\ds \size {\frac {e^{-\lambda^2 \paren {1 + x^2} } } {1 + x^2} } \le \frac 1 {1 + x^2}$
for each $x \in \R$.
Note that from [[Definite Integral to Infinity of Reciprocal of x Squared plus a Squared|Definite Integ... | Integral to Infinity of Exponential of -t^2/Proof 2 | https://proofwiki.org/wiki/Integral_to_Infinity_of_Exponential_of_-t^2 | https://proofwiki.org/wiki/Integral_to_Infinity_of_Exponential_of_-t^2/Proof_2 | [
"Integral to Infinity of Exponential of -t^2",
"Gauss Error Function",
"Definite Integrals involving Exponential Function"
] | [] | [
"Definition:Positive/Real Number",
"Definite Integral to Infinity of Reciprocal of x Squared plus a Squared",
"Comparison Test for Improper Integral",
"Definition:Real Function",
"Definite Integral of Partial Derivative",
"Derivative of Exponential Function",
"Integration by Substitution",
"Definite I... |
proofwiki-15916 | Laplace Transform of Real Power | Let $n$ be a constant real number such that $n > -1$
Let $f: \R \to \R$ be the real function defined as:
:$\map f t = t^n$
Then $f$ has a Laplace transform given by:
{{begin-eqn}}
{{eqn | l = \laptrans {\map f t}
| r = \int_0^\infty e^{-s t} t^n \rd t
}}
{{eqn | r = \frac {\map \Gamma {n + 1} } {s^{n + 1} }
}}
{{... | {{begin-eqn}}
{{eqn | l = \laptrans {t^n}
| r = \int_0^\infty e^{-s t} t^n \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^\infty e^{-u} \paren {\dfrac u s}^n \rd \paren {\dfrac u s}
| c = Integration by Substitution: $u := s t$ where $s > 0$ is assumed
}}
{{eqn | r = \dfrac 1 {s^{n + 1}... | Let $n$ be a [[Definition:Constant|constant]] [[Definition:Real Number|real number]] such that $n > -1$
Let $f: \R \to \R$ be the [[Definition:Real Function|real function]] defined as:
:$\map f t = t^n$
Then $f$ has a [[Definition:Laplace Transform|Laplace transform]] given by:
{{begin-eqn}}
{{eqn | l = \laptrans {... | {{begin-eqn}}
{{eqn | l = \laptrans {t^n}
| r = \int_0^\infty e^{-s t} t^n \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_0^\infty e^{-u} \paren {\dfrac u s}^n \rd \paren {\dfrac u s}
| c = [[Integration by Substitution]]: $u := s t$ where $s > 0$ is assumed
}}
{{eqn | r = \dfrac 1 {s^{n ... | Laplace Transform of Real Power | https://proofwiki.org/wiki/Laplace_Transform_of_Real_Power | https://proofwiki.org/wiki/Laplace_Transform_of_Real_Power | [
"Laplace Transform of Real Power",
"Examples of Laplace Transforms",
"Definite Integrals involving Exponential Function"
] | [
"Definition:Constant",
"Definition:Real Number",
"Definition:Real Function",
"Definition:Laplace Transform",
"Definition:Gamma Function"
] | [
"Integration by Substitution"
] |
proofwiki-15917 | Laplace Transform of Reciprocal of Square Root | :$\forall t \in \R_{\ne 0}: \laptrans {\dfrac 1 {\sqrt t} } = \sqrt {\dfrac \pi s}$
where $\laptrans f$ denotes the Laplace transform of the real function $f$. | Let $\map f t = \dfrac 1 {\sqrt t}$.
By definition of the Laplace transform with a discontinuity at zero, $\laptrans f$ is the improper integral:
:$\ds \laptrans f := \lim_{\epsilon \mathop \to 0^+} \int_\epsilon^{\to +\infty} \dfrac {e^{-s t} } {\sqrt t} \rd t$
if it exists.
{{explain|Demonstrate that it does exist}}
... | :$\forall t \in \R_{\ne 0}: \laptrans {\dfrac 1 {\sqrt t} } = \sqrt {\dfrac \pi s}$
where $\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|real function]] $f$. | Let $\map f t = \dfrac 1 {\sqrt t}$.
By definition of [[Definition:Laplace Transform/Discontinuity at Zero|the Laplace transform with a discontinuity at zero]], $\laptrans f$ is the [[Definition:Improper Integral|improper integral]]:
:$\ds \laptrans f := \lim_{\epsilon \mathop \to 0^+} \int_\epsilon^{\to +\infty} \dfr... | Laplace Transform of Reciprocal of Square Root | https://proofwiki.org/wiki/Laplace_Transform_of_Reciprocal_of_Square_Root | https://proofwiki.org/wiki/Laplace_Transform_of_Reciprocal_of_Square_Root | [
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function"
] | [
"Definition:Laplace Transform/Discontinuity at Zero",
"Definition:Improper Integral",
"Definition:Improper Integral",
"Definition:Convergent Integral",
"Definition:Laplace Transform",
"Laplace Transform of Real Power",
"Gamma Function of One Half"
] |
proofwiki-15918 | Gamma Function of 3 over 2 | :$\map \Gamma {\dfrac 3 2} = \dfrac {\sqrt \pi} 2$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {\dfrac 3 2}
| r = \map \Gamma {\dfrac 1 2 + 1}
| c =
}}
{{eqn | r = \dfrac 1 2 \map \Gamma {\dfrac 1 2}
| c = Gamma Difference Equation
}}
{{eqn | r = \dfrac {\sqrt \pi} 2
| c = Gamma Function of One Half
}}
{{end-eqn}}
{{qed}}
Category:Examples of Gamma F... | :$\map \Gamma {\dfrac 3 2} = \dfrac {\sqrt \pi} 2$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {\dfrac 3 2}
| r = \map \Gamma {\dfrac 1 2 + 1}
| c =
}}
{{eqn | r = \dfrac 1 2 \map \Gamma {\dfrac 1 2}
| c = [[Gamma Difference Equation]]
}}
{{eqn | r = \dfrac {\sqrt \pi} 2
| c = [[Gamma Function of One Half]]
}}
{{end-eqn}}
{{qed}}
[[Category:Examples... | Gamma Function of 3 over 2 | https://proofwiki.org/wiki/Gamma_Function_of_3_over_2 | https://proofwiki.org/wiki/Gamma_Function_of_3_over_2 | [
"Examples of Gamma Function Values"
] | [] | [
"Gamma Difference Equation",
"Gamma Function of One Half",
"Category:Examples of Gamma Function Values"
] |
proofwiki-15919 | Gamma Function of Minus 3 over 2 | :$\map \Gamma {-\dfrac 3 2} = \dfrac {4 \sqrt \pi} 3$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {-\dfrac 1 2}
| r = -\dfrac 3 2 \map \Gamma {-\dfrac 3 2}
| c = Gamma Difference Equation
}}
{{eqn | ll= \leadsto
| l = \map \Gamma {-\dfrac 3 2}
| r = -\dfrac 2 3 \map \Gamma {-\dfrac 1 2}
| c =
}}
{{eqn | r = -\dfrac 2 3 \paren {-2 \sqrt \pi}
... | :$\map \Gamma {-\dfrac 3 2} = \dfrac {4 \sqrt \pi} 3$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {-\dfrac 1 2}
| r = -\dfrac 3 2 \map \Gamma {-\dfrac 3 2}
| c = [[Gamma Difference Equation]]
}}
{{eqn | ll= \leadsto
| l = \map \Gamma {-\dfrac 3 2}
| r = -\dfrac 2 3 \map \Gamma {-\dfrac 1 2}
| c =
}}
{{eqn | r = -\dfrac 2 3 \paren {-2 \sqrt \pi}
... | Gamma Function of Minus 3 over 2 | https://proofwiki.org/wiki/Gamma_Function_of_Minus_3_over_2 | https://proofwiki.org/wiki/Gamma_Function_of_Minus_3_over_2 | [
"Examples of Gamma Function Values"
] | [] | [
"Gamma Difference Equation",
"Gamma Function of Minus One Half"
] |
proofwiki-15920 | Gamma Function of Minus 5 over 2 | :$\map \Gamma {-\dfrac 5 2} = -\dfrac {8 \sqrt \pi} {15}$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {-\dfrac 3 2}
| r = -\dfrac 5 2 \map \Gamma {-\dfrac 5 2}
| c = Gamma Difference Equation
}}
{{eqn | ll= \leadsto
| l = \map \Gamma {-\dfrac 5 2}
| r = -\dfrac 2 5 \map \Gamma {-\dfrac 3 2}
| c =
}}
{{eqn | r = -\dfrac 2 5 \paren {\dfrac {4 \sqrt \pi}... | :$\map \Gamma {-\dfrac 5 2} = -\dfrac {8 \sqrt \pi} {15}$ | {{begin-eqn}}
{{eqn | l = \map \Gamma {-\dfrac 3 2}
| r = -\dfrac 5 2 \map \Gamma {-\dfrac 5 2}
| c = [[Gamma Difference Equation]]
}}
{{eqn | ll= \leadsto
| l = \map \Gamma {-\dfrac 5 2}
| r = -\dfrac 2 5 \map \Gamma {-\dfrac 3 2}
| c =
}}
{{eqn | r = -\dfrac 2 5 \paren {\dfrac {4 \sqrt ... | Gamma Function of Minus 5 over 2 | https://proofwiki.org/wiki/Gamma_Function_of_Minus_5_over_2 | https://proofwiki.org/wiki/Gamma_Function_of_Minus_5_over_2 | [
"Examples of Gamma Function Values"
] | [] | [
"Gamma Difference Equation",
"Gamma Function of Minus 3 over 2"
] |
proofwiki-15921 | Gamma Function of Zero | :$\map \Gamma 0$ is not defined. | {{begin-eqn}}
{{eqn | l = \map \Gamma 1
| r = 0 \, \map \Gamma 0
| c = Gamma Difference Equation
}}
{{eqn | ll= \leadsto
| l = \map \Gamma 0
| r = \dfrac {\map \Gamma 1} 0
| c =
}}
{{eqn | r = \dfrac 1 0
| c = Gamma Function Extends Factorial
}}
{{end-eqn}}
But $\dfrac 1 0$ is not d... | :$\map \Gamma 0$ is not defined. | {{begin-eqn}}
{{eqn | l = \map \Gamma 1
| r = 0 \, \map \Gamma 0
| c = [[Gamma Difference Equation]]
}}
{{eqn | ll= \leadsto
| l = \map \Gamma 0
| r = \dfrac {\map \Gamma 1} 0
| c =
}}
{{eqn | r = \dfrac 1 0
| c = [[Gamma Function Extends Factorial]]
}}
{{end-eqn}}
But $\dfrac 1 0$... | Gamma Function of Zero | https://proofwiki.org/wiki/Gamma_Function_of_Zero | https://proofwiki.org/wiki/Gamma_Function_of_Zero | [
"Examples of Gamma Function Values"
] | [] | [
"Gamma Difference Equation",
"Gamma Function Extends Factorial"
] |
proofwiki-15922 | Gamma Function of Minus 1 | :$\map \Gamma {-1}$ is not defined. | {{begin-eqn}}
{{eqn | l = \map \Gamma 0
| r = \paren {-1} \, \map \Gamma {-1}
| c = Gamma Difference Equation
}}
{{eqn | ll= \leadsto
| l = \map \Gamma {-1}
| r = \dfrac {\map \Gamma 0} {-1}
| c =
}}
{{end-eqn}}
But from Gamma Function of Zero, $\map \Gamma 0$ is not defined.
Hence the re... | :$\map \Gamma {-1}$ is not defined. | {{begin-eqn}}
{{eqn | l = \map \Gamma 0
| r = \paren {-1} \, \map \Gamma {-1}
| c = [[Gamma Difference Equation]]
}}
{{eqn | ll= \leadsto
| l = \map \Gamma {-1}
| r = \dfrac {\map \Gamma 0} {-1}
| c =
}}
{{end-eqn}}
But from [[Gamma Function of Zero]], $\map \Gamma 0$ is not defined.
He... | Gamma Function of Minus 1 | https://proofwiki.org/wiki/Gamma_Function_of_Minus_1 | https://proofwiki.org/wiki/Gamma_Function_of_Minus_1 | [
"Examples of Gamma Function Values"
] | [] | [
"Gamma Difference Equation",
"Gamma Function of Zero"
] |
proofwiki-15923 | Gamma Function of Minus 2 | :$\map \Gamma {-2}$ is not defined. | {{begin-eqn}}
{{eqn | l = \map \Gamma {-1}
| r = \paren {-2} \, \map \Gamma {-2}
| c = Gamma Difference Equation
}}
{{eqn | ll= \leadsto
| l = \map \Gamma {-2}
| r = \dfrac {\map \Gamma {-1} } {-2}
| c =
}}
{{end-eqn}}
But from Gamma Function of Minus 1, $\map \Gamma {-1}$ is not defined.... | :$\map \Gamma {-2}$ is not defined. | {{begin-eqn}}
{{eqn | l = \map \Gamma {-1}
| r = \paren {-2} \, \map \Gamma {-2}
| c = [[Gamma Difference Equation]]
}}
{{eqn | ll= \leadsto
| l = \map \Gamma {-2}
| r = \dfrac {\map \Gamma {-1} } {-2}
| c =
}}
{{end-eqn}}
But from [[Gamma Function of Minus 1]], $\map \Gamma {-1}$ is not... | Gamma Function of Minus 2 | https://proofwiki.org/wiki/Gamma_Function_of_Minus_2 | https://proofwiki.org/wiki/Gamma_Function_of_Minus_2 | [
"Examples of Gamma Function Values"
] | [] | [
"Gamma Difference Equation",
"Gamma Function of Minus 1"
] |
proofwiki-15924 | Integral to Infinity of Bessel Function of First Kind order Zero | :$\ds \int_0^\infty \map {J_0} t \rd t = 1$
where $J_0$ denotes the Bessel function of the first kind of order $0$. | Using the technique of Evaluation of Integral using Laplace Transform:
{{begin-eqn}}
{{eqn | l = \int_0^\infty e^{-s t} \map {J_0} t \rd t
| r = \dfrac 1 {\sqrt {s^2 + 1} }
| c = Laplace Transform of Bessel Function of the First Kind of Order Zero
}}
{{eqn | ll= \leadsto
| l = \int_0^\infty \map {J_0}... | :$\ds \int_0^\infty \map {J_0} t \rd t = 1$
where $J_0$ denotes the [[Definition:Bessel Function of the First Kind|Bessel function of the first kind]] of [[Definition:Order of Bessel Function|order $0$]]. | Using the technique of [[Evaluation of Integral using Laplace Transform]]:
{{begin-eqn}}
{{eqn | l = \int_0^\infty e^{-s t} \map {J_0} t \rd t
| r = \dfrac 1 {\sqrt {s^2 + 1} }
| c = [[Laplace Transform of Bessel Function of the First Kind of Order Zero]]
}}
{{eqn | ll= \leadsto
| l = \int_0^\infty \... | Integral to Infinity of Bessel Function of First Kind order Zero | https://proofwiki.org/wiki/Integral_to_Infinity_of_Bessel_Function_of_First_Kind_order_Zero | https://proofwiki.org/wiki/Integral_to_Infinity_of_Bessel_Function_of_First_Kind_order_Zero | [
"Bessel Functions"
] | [
"Definition:Bessel Function/First Kind",
"Definition:Bessel Function/Order"
] | [
"Evaluation of Integral using Laplace Transform",
"Laplace Transform of Bessel Function of the First Kind of Order Zero"
] |
proofwiki-15925 | Integral to Infinity of e^-t by Gauss Error Function of Root t | :$\ds \int_0^\infty e^{-t} \erf \sqrt t \rd t = \dfrac {\sqrt 2} 2$ | Using the technique of Evaluation of Integral using Laplace Transform:
{{begin-eqn}}
{{eqn | l = \int_0^\infty e^{-s t} \erf \sqrt t \rd t
| r = \dfrac 1 {s \sqrt {s + 1} }
| c = Laplace Transform of Gauss Error Function of Root
}}
{{eqn | ll= \leadsto
| l = \int_0^\infty \erf \sqrt t \rd t
| r ... | :$\ds \int_0^\infty e^{-t} \erf \sqrt t \rd t = \dfrac {\sqrt 2} 2$ | Using the technique of [[Evaluation of Integral using Laplace Transform]]:
{{begin-eqn}}
{{eqn | l = \int_0^\infty e^{-s t} \erf \sqrt t \rd t
| r = \dfrac 1 {s \sqrt {s + 1} }
| c = [[Laplace Transform of Gauss Error Function of Root]]
}}
{{eqn | ll= \leadsto
| l = \int_0^\infty \erf \sqrt t \rd t
... | Integral to Infinity of e^-t by Gauss Error Function of Root t | https://proofwiki.org/wiki/Integral_to_Infinity_of_e^-t_by_Gauss_Error_Function_of_Root_t | https://proofwiki.org/wiki/Integral_to_Infinity_of_e^-t_by_Gauss_Error_Function_of_Root_t | [
"Gauss Error Function",
"Definite Integrals involving Exponential Function"
] | [] | [
"Evaluation of Integral using Laplace Transform",
"Laplace Transform of Gauss Error Function of Root"
] |
proofwiki-15926 | Laplace Transform of Natural Logarithm | :$\laptrans {\ln t} = \dfrac {\map {\Gamma'} 1 - \ln s} s = -\dfrac {\gamma + \ln s} s$
where:
:$\laptrans f$ denotes the Laplace transform of the function $f$
:$\Gamma$ denotes the Gamma function
:$\gamma$ denotes the Euler-Mascheroni constant. | From Laplace Transform of Power:
:$\ds \int_0^\infty e^{-s t} t^k \rd t = \dfrac {\map \Gamma {k + 1} } {s^{k + 1} }$
for $k > -1$.
Differentiating {{WRT|Differentiation}} $k$:
:$\ds \int_0^\infty e^{-s t} t^k \ln t \rd t = \dfrac {\map {\Gamma'} {k + 1} - \map \Gamma {k + 1} \ln s} {s^{k + 1} }$
Setting $k = 0$:
{{beg... | :$\laptrans {\ln t} = \dfrac {\map {\Gamma'} 1 - \ln s} s = -\dfrac {\gamma + \ln s} s$
where:
:$\laptrans f$ denotes the [[Definition:Laplace Transform|Laplace transform]] of the [[Definition:Real Function|function]] $f$
:$\Gamma$ denotes the [[Definition:Gamma Function|Gamma function]]
:$\gamma$ denotes the [[Defini... | From [[Laplace Transform of Power]]:
:$\ds \int_0^\infty e^{-s t} t^k \rd t = \dfrac {\map \Gamma {k + 1} } {s^{k + 1} }$
for $k > -1$.
[[Definition:Differentiation|Differentiating]] {{WRT|Differentiation}} $k$:
:$\ds \int_0^\infty e^{-s t} t^k \ln t \rd t = \dfrac {\map {\Gamma'} {k + 1} - \map \Gamma {k + 1} \ln s... | Laplace Transform of Natural Logarithm/Proof 2 | https://proofwiki.org/wiki/Laplace_Transform_of_Natural_Logarithm | https://proofwiki.org/wiki/Laplace_Transform_of_Natural_Logarithm/Proof_2 | [
"Laplace Transform of Natural Logarithm",
"Natural Logarithms",
"Examples of Laplace Transforms"
] | [
"Definition:Laplace Transform",
"Definition:Real Function",
"Definition:Gamma Function",
"Definition:Euler-Mascheroni Constant"
] | [
"Laplace Transform of Power",
"Definition:Differentiation",
"Derivative of Gamma Function at 1"
] |
proofwiki-15927 | Second Derivative of Laplace Transform | Let $f: \R \to \R$ or $\R \to \C$ be a continuous function, twice differentiable on any closed interval $\closedint 0 a$.
Let $\laptrans f = F$ denote the Laplace transform of $f$.
Then, everywhere that $\dfrac {\d^2} {\d s^2} \laptrans f$ exists:
:$\dfrac {\d^2} {\d s^2} \laptrans {\map f t} = \laptrans {t^2 \, \map f... | {{begin-eqn}}
{{eqn | l = \dfrac {\d^2} {\d s^2} \laptrans {\map f t}
| r = \map {\frac \d {\d s} } {\dfrac \d {\d s} \laptrans {\map f t} }
| c = {{Defof|Second Derivative}}
}}
{{eqn | r = \map {\frac \d {\d s} } {-\laptrans {t \, \map f t} }
| c = Derivative of Laplace Transform
}}
{{eqn | r = -\fra... | Let $f: \R \to \R$ or $\R \to \C$ be a [[Definition:Continuous|continuous]] [[Definition:Function|function]], twice [[Definition:Differentiable on Interval|differentiable]] on any [[Definition:Closed Real Interval|closed interval]] $\closedint 0 a$.
Let $\laptrans f = F$ denote the [[Definition:Laplace Transform|Lapla... | {{begin-eqn}}
{{eqn | l = \dfrac {\d^2} {\d s^2} \laptrans {\map f t}
| r = \map {\frac \d {\d s} } {\dfrac \d {\d s} \laptrans {\map f t} }
| c = {{Defof|Second Derivative}}
}}
{{eqn | r = \map {\frac \d {\d s} } {-\laptrans {t \, \map f t} }
| c = [[Derivative of Laplace Transform]]
}}
{{eqn | r = -... | Second Derivative of Laplace Transform | https://proofwiki.org/wiki/Second_Derivative_of_Laplace_Transform | https://proofwiki.org/wiki/Second_Derivative_of_Laplace_Transform | [
"Derivatives of Laplace Transforms",
"Laplace Transforms",
"Derivatives"
] | [
"Definition:Continuous",
"Definition:Function",
"Definition:Differentiable Mapping/Real Function/Interval",
"Definition:Real Interval/Closed",
"Definition:Laplace Transform"
] | [
"Derivative of Laplace Transform",
"Derivative of Laplace Transform"
] |
proofwiki-15928 | Laplace Transform of Multiple Integral | :$\ds \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$n$ times} } \map f u \rd u^n} = \dfrac {\map F s} {s^n}$
wherever $\laptrans f$ exists. | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$n$ times} } \map f u \rd u^n} = \dfrac {\map F s} {s^n}$
$\map P 0$ is the case:
:$\map f u = \map F s$
which is the statement of the Laplace transform.
Thus $... | :$\ds \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$n$ times} } \map f u \rd u^n} = \dfrac {\map F s} {s^n}$
wherever $\laptrans f$ exists. | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \laptrans {\underbrace {\int_0^t \dotsm \int_0^t}_{\text {$n$ times} } \map f u \rd u^n} = \dfrac {\map F s} {s^n}$
$\map P 0$ is the case:
:$\map f... | Laplace Transform of Multiple Integral | https://proofwiki.org/wiki/Laplace_Transform_of_Multiple_Integral | https://proofwiki.org/wiki/Laplace_Transform_of_Multiple_Integral | [
"Laplace Transforms of Integrals",
"Laplace Transforms"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Laplace Transform",
"Principle of Mathematical Induction"
] |
proofwiki-15929 | Convolution Theorem | Let $\GF \in \set {\R, \C}$.
Let $f: \R \to \GF$ and $g: \R \to \GF$ be functions.
Let their Laplace transforms $\laptrans {\map f t} = \map F s$ and $\laptrans {\map g t} = \map G s$ exist.
Then:
:$\map F s \map G s = \ds \laptrans {\int_0^t \map f u \map g {t - u} \rd u}$ | {{begin-eqn}}
{{eqn | l = \laptrans {\int_0^t \map f u \map g {t - u} \rd u}
| r = \int_{t \mathop = 0}^\infty e^{-s t} \paren {\int_{u \mathop = 0}^t \map f u \map g {t - u} \rd u} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_{t \mathop = 0}^\infty \int_{u \mathop = 0}^t e^{-s t} \map f u \m... | Let $\GF \in \set {\R, \C}$.
Let $f: \R \to \GF$ and $g: \R \to \GF$ be [[Definition:Function|functions]].
Let their [[Definition:Laplace Transform|Laplace transforms]] $\laptrans {\map f t} = \map F s$ and $\laptrans {\map g t} = \map G s$ exist.
Then:
:$\map F s \map G s = \ds \laptrans {\int_0^t \map f u \map g... | {{begin-eqn}}
{{eqn | l = \laptrans {\int_0^t \map f u \map g {t - u} \rd u}
| r = \int_{t \mathop = 0}^\infty e^{-s t} \paren {\int_{u \mathop = 0}^t \map f u \map g {t - u} \rd u} \rd t
| c = {{Defof|Laplace Transform}}
}}
{{eqn | r = \int_{t \mathop = 0}^\infty \int_{u \mathop = 0}^t e^{-s t} \map f u \m... | Convolution Theorem/Proof 1 | https://proofwiki.org/wiki/Convolution_Theorem | https://proofwiki.org/wiki/Convolution_Theorem/Proof_1 | [
"Convolution Theorem",
"Laplace Transforms",
"Named Theorems"
] | [
"Definition:Function",
"Definition:Laplace Transform"
] | [
"Definition:Primitive (Calculus)/Integration",
"File:ConvolutionTheorem1.png",
"File:ConvolutionTheorem2.png",
"Change of Variables Theorem (Multivariable Calculus)",
"Definition:Jacobian",
"Definition:Function",
"Definition:Function",
"Definition:Quadrilateral/Square",
"File:ConvolutionTheorem3.png... |
proofwiki-15930 | Integral of Reciprocal is Divergent/Unbounded Above | :$\ds \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$ | From Harmonic Series is Divergent, we have that $\ds \sum_{n \mathop = 1}^\infty \frac 1 n$ diverges to $+\infty$.
Thus from the Cauchy Integral Test:
:$\ds \int_1^n \frac {\d x} x \to +\infty$
diverges.
{{qed}} | :$\ds \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$ | From [[Harmonic Series is Divergent]], we have that $\ds \sum_{n \mathop = 1}^\infty \frac 1 n$ [[Definition:Divergent Series|diverges]] to $+\infty$.
Thus from the [[Cauchy Integral Test]]:
:$\ds \int_1^n \frac {\d x} x \to +\infty$
[[Definition:Divergent Improper Integral|diverges]].
{{qed}} | Integral of Reciprocal is Divergent/Unbounded Above/Proof 1 | https://proofwiki.org/wiki/Integral_of_Reciprocal_is_Divergent/Unbounded_Above | https://proofwiki.org/wiki/Integral_of_Reciprocal_is_Divergent/Unbounded_Above/Proof_1 | [
"Integral of Reciprocal is Divergent"
] | [] | [
"Harmonic Series is Divergent",
"Definition:Divergent Series",
"Cauchy Integral Test",
"Definition:Divergent Improper Integral"
] |
proofwiki-15931 | Integral of Reciprocal is Divergent/Unbounded Above | :$\ds \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$ | From the definition of natural logarithm:
:$\ds \ln x = \int_1^x \dfrac 1 t \rd t$
The result follows from Logarithm Tends to Infinity.
{{qed}} | :$\ds \int_1^n \frac {\d x} x \to +\infty$ as $n \to + \infty$ | From the definition of [[Definition:Real Natural Logarithm|natural logarithm]]:
:$\ds \ln x = \int_1^x \dfrac 1 t \rd t$
The result follows from [[Logarithm Tends to Infinity]].
{{qed}} | Integral of Reciprocal is Divergent/Unbounded Above/Proof 2 | https://proofwiki.org/wiki/Integral_of_Reciprocal_is_Divergent/Unbounded_Above | https://proofwiki.org/wiki/Integral_of_Reciprocal_is_Divergent/Unbounded_Above/Proof_2 | [
"Integral of Reciprocal is Divergent"
] | [] | [
"Definition:Natural Logarithm/Positive Real",
"Logarithm Tends to Infinity"
] |
proofwiki-15932 | Integral of Reciprocal is Divergent/To Zero | :$\ds \int_\gamma^1 \frac {\d x} x \to -\infty$ as $\gamma \to 0^+$ | Put $x = \dfrac 1 z$.
Then:
{{begin-eqn}}
{{eqn | l = \int_\gamma^1 \frac {\d x} x
| r = \int_{1 / \gamma}^1 \frac {-z} {z^2} \rd z
| c = Integration by Substitution
}}
{{eqn | r = \int_1^{1 / \gamma} \frac {\d z} z
| c =
}}
{{end-eqn}}
But as $\gamma \to 0+$, we have that $\dfrac 1 \gamma \to +\inft... | :$\ds \int_\gamma^1 \frac {\d x} x \to -\infty$ as $\gamma \to 0^+$ | Put $x = \dfrac 1 z$.
Then:
{{begin-eqn}}
{{eqn | l = \int_\gamma^1 \frac {\d x} x
| r = \int_{1 / \gamma}^1 \frac {-z} {z^2} \rd z
| c = [[Integration by Substitution]]
}}
{{eqn | r = \int_1^{1 / \gamma} \frac {\d z} z
| c =
}}
{{end-eqn}}
But as $\gamma \to 0+$, we have that $\dfrac 1 \gamma \t... | Integral of Reciprocal is Divergent/To Zero | https://proofwiki.org/wiki/Integral_of_Reciprocal_is_Divergent/To_Zero | https://proofwiki.org/wiki/Integral_of_Reciprocal_is_Divergent/To_Zero | [
"Integral of Reciprocal is Divergent"
] | [] | [
"Integration by Substitution",
"Integral of Reciprocal is Divergent/Unbounded Above"
] |
proofwiki-15933 | Convergence of P-Series/Absolute Convergence if Real Part of p Greater than 1 | Let $\map \Re p > 1$.
Then the $p$-series:
:$\ds \sum_{n \mathop = 1}^\infty n^{-p}$
converges absolutely. | === Lemma ===
{{:Convergence of P-Series/Lemma}}
Since $x > 1$ it follows that $1 - x < 0$.
Thus $P^{1 - x} \to 0$ as $P \to \infty$.
Setting $x - 1 = \delta >0$, this limit is:
:$\ds -\frac 1 {\delta} \lim_{t \mathop \to \infty} \frac 1 {t^\delta} = 0$
Hence the integral is just $\dfrac 1 {1 - x}$ (that is, convergen... | Let $\map \Re p > 1$.
Then the [[Definition:P-Series|$p$-series]]:
:$\ds \sum_{n \mathop = 1}^\infty n^{-p}$
[[Definition:Absolutely Convergent Series|converges absolutely]]. | === [[Convergence of P-Series/Lemma|Lemma]] ===
{{:Convergence of P-Series/Lemma}}
Since $x > 1$ it follows that $1 - x < 0$.
Thus $P^{1 - x} \to 0$ as $P \to \infty$.
Setting $x - 1 = \delta >0$, this [[Definition:Limit of Real Function|limit]] is:
:$\ds -\frac 1 {\delta} \lim_{t \mathop \to \infty} \frac 1 {t^\d... | Convergence of P-Series/Absolute Convergence if Real Part of p Greater than 1 | https://proofwiki.org/wiki/Convergence_of_P-Series/Absolute_Convergence_if_Real_Part_of_p_Greater_than_1 | https://proofwiki.org/wiki/Convergence_of_P-Series/Absolute_Convergence_if_Real_Part_of_p_Greater_than_1 | [
"Convergence of P-Series"
] | [
"Definition:P-Series",
"Definition:Absolutely Convergent Series"
] | [
"Convergence of P-Series/Lemma",
"Definition:Limit of Real Function"
] |
proofwiki-15934 | Convergence of P-Series/Divergence if p between 0 and 1 | Let $0 < \map \Re p \le 1$.
Then the $p$-series:
:$\ds \sum_{n \mathop = 1}^\infty n^{-p}$
diverges. | === Lemma ===
{{:Convergence of P-Series/Lemma}}{{qed|lemma}}
Hence, the convergence of the $p$-series is dependent on the convergence of:
:$\ds \lim_{t \mathop \to \infty} \frac {t^{1 - x} } {1 - x}$
Suppose $0 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \lim_{t \mathop \to \infty} \frac {t^{1 - x} } {1 - x}
| r ... | Let $0 < \map \Re p \le 1$.
Then the [[Definition:P-Series|$p$-series]]:
:$\ds \sum_{n \mathop = 1}^\infty n^{-p}$
[[Definition:Divergent Series|diverges]]. | === [[Convergence of P-Series/Lemma|Lemma]] ===
{{:Convergence of P-Series/Lemma}}{{qed|lemma}}
Hence, the [[Definition:Convergent Real Series|convergence]] of the [[Definition:P-Series|$p$-series]] is dependent on the [[Definition:Convergent Real Series|convergence]] of:
:$\ds \lim_{t \mathop \to \infty} \frac {t^... | Convergence of P-Series/Divergence if p between 0 and 1 | https://proofwiki.org/wiki/Convergence_of_P-Series/Divergence_if_p_between_0_and_1 | https://proofwiki.org/wiki/Convergence_of_P-Series/Divergence_if_p_between_0_and_1 | [
"Convergence of P-Series"
] | [
"Definition:P-Series",
"Definition:Divergent Series"
] | [
"Convergence of P-Series/Lemma",
"Definition:Convergent Series/Number Field",
"Definition:P-Series",
"Definition:Convergent Series/Number Field",
"Limit at Infinity of x^n",
"Integral of Reciprocal is Divergent",
"Cauchy Integral Test",
"Category:Convergence of P-Series"
] |
proofwiki-15935 | Integral to Infinity of Reciprocal of Power of x | The improper integral
:$\ds \int_1^\infty \dfrac {\d t} {t^x}$
exists {{iff}} $x > 1$. | First let $x \ne 1$.
Then:
{{begin-eqn}}
{{eqn | l = \int_1^\infty \dfrac {\d t} {t^x}
| r = \lim_{P \mathop \to \infty} \int_1^P t^{-x} \rd t
| c = {{Defof|Improper Integral}}
}}
{{eqn | r = \lim_{P \mathop \to \infty} \intlimits {\dfrac {t^{-x + 1} } {-x + 1} } 1 P
| c = Primitive of Power
}}
{{eqn ... | The [[Definition:Improper Integral|improper integral]]
:$\ds \int_1^\infty \dfrac {\d t} {t^x}$
exists {{iff}} $x > 1$. | First let $x \ne 1$.
Then:
{{begin-eqn}}
{{eqn | l = \int_1^\infty \dfrac {\d t} {t^x}
| r = \lim_{P \mathop \to \infty} \int_1^P t^{-x} \rd t
| c = {{Defof|Improper Integral}}
}}
{{eqn | r = \lim_{P \mathop \to \infty} \intlimits {\dfrac {t^{-x + 1} } {-x + 1} } 1 P
| c = [[Primitive of Power]]
}}
... | Integral to Infinity of Reciprocal of Power of x | https://proofwiki.org/wiki/Integral_to_Infinity_of_Reciprocal_of_Power_of_x | https://proofwiki.org/wiki/Integral_to_Infinity_of_Reciprocal_of_Power_of_x | [
"Reciprocals",
"Examples of Definite Integrals"
] | [
"Definition:Improper Integral"
] | [
"Primitive of Power",
"Sequence of Powers of Reciprocals is Null Sequence",
"Reciprocal of Null Sequence",
"Integral of Reciprocal is Divergent"
] |
proofwiki-15936 | Convergence of P-Series/Real | Let $p \in \R$ be a real number.
Then the $p$-series:
:$\ds \sum_{n \mathop = 1}^\infty n^{-p}$
is convergent {{iff}} $p > 1$. | By the Cauchy Integral Test:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^x}$ converges {{iff}} the improper integral $\ds \int_1^\infty \frac {\d t} {t^x}$ exists.
The result follows from Integral to Infinity of Reciprocal of Power of x.
{{qed}} | Let $p \in \R$ be a [[Definition:Real Number|real number]].
Then the [[Definition:P-Series|$p$-series]]:
:$\ds \sum_{n \mathop = 1}^\infty n^{-p}$
is [[Definition:Convergent Real Series|convergent]] {{iff}} $p > 1$. | By the [[Cauchy Integral Test]]:
:$\ds \sum_{n \mathop = 1}^\infty \frac 1 {n^x}$ [[Definition:Convergent Real Sequence|converges]] {{iff}} the [[Definition:Improper Integral|improper integral]] $\ds \int_1^\infty \frac {\d t} {t^x}$ exists.
The result follows from [[Integral to Infinity of Reciprocal of Power of x]]... | Convergence of P-Series/Real/Proof 1 | https://proofwiki.org/wiki/Convergence_of_P-Series/Real | https://proofwiki.org/wiki/Convergence_of_P-Series/Real/Proof_1 | [
"Convergence of P-Series"
] | [
"Definition:Real Number",
"Definition:P-Series",
"Definition:Convergent Series/Number Field"
] | [
"Cauchy Integral Test",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Improper Integral",
"Integral to Infinity of Reciprocal of Power of x"
] |
proofwiki-15937 | Convergence of P-Series/Real | Let $p \in \R$ be a real number.
Then the $p$-series:
:$\ds \sum_{n \mathop = 1}^\infty n^{-p}$
is convergent {{iff}} $p > 1$. | Let $p = 1$.
Then from Harmonic Series is Divergent the $p$-series diverges.
So let $p > 1$.
We note that the sequence of partial sums is increasing.
Hence it is sufficient to show that they are bounded above.
Let:
:$s_{2^N} := 1 + \dfrac 1 {2^p} + \dfrac 1 {3^p} + \dotsb + \dfrac 1 {N^p}$
Then:
{{begin-eqn}}
{{eqn | l... | Let $p \in \R$ be a [[Definition:Real Number|real number]].
Then the [[Definition:P-Series|$p$-series]]:
:$\ds \sum_{n \mathop = 1}^\infty n^{-p}$
is [[Definition:Convergent Real Series|convergent]] {{iff}} $p > 1$. | Let $p = 1$.
Then from [[Harmonic Series is Divergent]] the [[Definition:P-Series|$p$-series]] [[Definition:Divergent Series|diverges]].
So let $p > 1$.
We note that the [[Definition:Sequence of Partial Sums|sequence of partial sums]] is [[Definition:Increasing Real Sequence|increasing]].
Hence it is [[Definition:... | Convergence of P-Series/Real/Proof 2 | https://proofwiki.org/wiki/Convergence_of_P-Series/Real | https://proofwiki.org/wiki/Convergence_of_P-Series/Real/Proof_2 | [
"Convergence of P-Series"
] | [
"Definition:Real Number",
"Definition:P-Series",
"Definition:Convergent Series/Number Field"
] | [
"Harmonic Series is Divergent",
"Definition:P-Series",
"Definition:Divergent Series",
"Definition:Series/Sequence of Partial Sums",
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Conditional/Sufficient Condition",
"Definition:Bounded Above Sequence/Real"
] |
proofwiki-15938 | Transitive Law | Let $a, b, c \in \R$ such that $a > b$ and $b > c$.
Then:
:$a > c$ | From Real Numbers form Ordered Integral Domain, $\struct {\R, +, \times, \le}$ forms an ordered integral domain.
From Ordered Integral Domain is Totally Ordered Ring, the usual ordering $\le$ is a total ordering.
From Relation Induced by Strict Positivity Property is Transitive it follows that $<$ is transitive.
{{qed}... | Let $a, b, c \in \R$ such that $a > b$ and $b > c$.
Then:
:$a > c$ | From [[Real Numbers form Ordered Integral Domain]], $\struct {\R, +, \times, \le}$ forms an [[Definition:Ordered Integral Domain|ordered integral domain]].
From [[Ordered Integral Domain is Totally Ordered Ring]], the [[Definition:Usual Ordering|usual ordering]] $\le$ is a [[Definition:Total Ordering|total ordering]].... | Transitive Law | https://proofwiki.org/wiki/Transitive_Law | https://proofwiki.org/wiki/Transitive_Law | [
"Transitive Law",
"Real Numbers",
"Inequalities",
"Named Theorems"
] | [] | [
"Real Numbers form Ordered Integral Domain",
"Definition:Ordered Integral Domain",
"Ordered Integral Domain is Totally Ordered Ring",
"Definition:Usual Ordering",
"Definition:Total Ordering",
"Relation Induced by Strict Positivity Property is Transitive",
"Definition:Transitive Relation"
] |
proofwiki-15939 | Real Number Ordering is Compatible with Multiplication/Positive Factor | :$\forall a, b, c \in \R: a < b \land c > 0 \implies a c < b c$ | From Real Numbers form Ordered Integral Domain, $\struct {\R, +, \times, \le}$ forms an ordered integral domain.
Thus:
{{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = b - a
| o = >
| r = 0
| c = {{Defof|Positivity Property}}
}}
{{eqn | ll= \lead... | :$\forall a, b, c \in \R: a < b \land c > 0 \implies a c < b c$ | From [[Real Numbers form Ordered Integral Domain]], $\struct {\R, +, \times, \le}$ forms an [[Definition:Ordered Integral Domain|ordered integral domain]].
Thus:
{{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = b - a
| o = >
| r = 0
| c = {{Def... | Real Number Ordering is Compatible with Multiplication/Positive Factor | https://proofwiki.org/wiki/Real_Number_Ordering_is_Compatible_with_Multiplication/Positive_Factor | https://proofwiki.org/wiki/Real_Number_Ordering_is_Compatible_with_Multiplication/Positive_Factor | [
"Real Number Ordering is Compatible with Multiplication"
] | [] | [
"Real Numbers form Ordered Integral Domain",
"Definition:Ordered Integral Domain",
"Definition:Closed under Mapping",
"Definition:Ring (Abstract Algebra)/Addition"
] |
proofwiki-15940 | Real Number Ordering is Compatible with Multiplication/Negative Factor | :$\forall a, b, c \in \R: a < b \land c < 0 \implies a c > b c$ | From Real Numbers form Ordered Integral Domain, $\struct {\R, +, \times, \le}$ forms an ordered integral domain.
Thus:
{{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = b - a
| o = >
| r = 0
| c = {{Defof|Positivity Property}}
}}
{{eqn | ll= \lead... | :$\forall a, b, c \in \R: a < b \land c < 0 \implies a c > b c$ | From [[Real Numbers form Ordered Integral Domain]], $\struct {\R, +, \times, \le}$ forms an [[Definition:Ordered Integral Domain|ordered integral domain]].
Thus:
{{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = b - a
| o = >
| r = 0
| c = {{Def... | Real Number Ordering is Compatible with Multiplication/Negative Factor | https://proofwiki.org/wiki/Real_Number_Ordering_is_Compatible_with_Multiplication/Negative_Factor | https://proofwiki.org/wiki/Real_Number_Ordering_is_Compatible_with_Multiplication/Negative_Factor | [
"Real Number Ordering is Compatible with Multiplication"
] | [] | [
"Real Numbers form Ordered Integral Domain",
"Definition:Ordered Integral Domain",
"Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative"
] |
proofwiki-15941 | Rational Power of Product of Real Numbers | Let $r, s \in \R_{> 0}$ be (strictly) positive real numbers.
<onlyinclude>
Let $x \in \Q$ be a rational number.
Let $r^x$ be defined as $r$ to the power of $x$.
Then:
:$\paren {r s}^x = r^x s^x$ | Let $x = \dfrac p q$ where $p, q \in \Z$ and $q > 0$.
We have:
{{begin-eqn}}
{{eqn | l = r^x s^x
| r = \paren {r^p}^{1 / q} \paren {s^p}^{1 / q}
| c =
}}
{{eqn | r = \paren {r^p s^p}^{1 / q}
| c =
}}
{{eqn | r = \paren {\paren {r s}^p}^{1 / q}
| c =
}}
{{eqn | r = \paren {r s}^{p / q}
|... | Let $r, s \in \R_{> 0}$ be [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]].
<onlyinclude>
Let $x \in \Q$ be a [[Definition:Rational Number|rational number]].
Let $r^x$ be defined as [[Definition:Rational Power|$r$ to the power of $x$]].
Then:
:$\paren {r s}^x = r^x s^x$ | Let $x = \dfrac p q$ where $p, q \in \Z$ and $q > 0$.
We have:
{{begin-eqn}}
{{eqn | l = r^x s^x
| r = \paren {r^p}^{1 / q} \paren {s^p}^{1 / q}
| c =
}}
{{eqn | r = \paren {r^p s^p}^{1 / q}
| c =
}}
{{eqn | r = \paren {\paren {r s}^p}^{1 / q}
| c =
}}
{{eqn | r = \paren {r s}^{p / q}
... | Rational Power of Product of Real Numbers | https://proofwiki.org/wiki/Rational_Power_of_Product_of_Real_Numbers | https://proofwiki.org/wiki/Rational_Power_of_Product_of_Real_Numbers | [
"Powers"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Rational Number",
"Definition:Power (Algebra)/Rational Number"
] | [] |
proofwiki-15942 | Sign of Quadratic Function Between Roots | Let $a \in \R_{>0}$ be a (strictly) positive real number.
Let $\alpha$ and $\beta$, where $\alpha < \beta$, be the roots of the quadratic function:
:$\map Q x = a x^2 + b x + c$
whose discriminant $b^2 - 4 a c$ is (strictly) positive.
Then:
:$\begin {cases} \map Q x < 0 & : \text {when $\alpha < x < \beta$} \\ \map Q x... | Because $b^2 - 4 a c > 0$, we have from Solution to Quadratic Equation with Real Coefficients that the roots of $\map Q x$ are real and unequal.
This demonstrates the existence of $\alpha$ and $\beta$, where by hypothesis we state that $\alpha < \beta$.
We can express $\map Q x$ as:
:$\map Q x = a \paren {x - \alpha} \... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Let $\alpha$ and $\beta$, where $\alpha < \beta$, be the [[Definition:Root of Polynomial|roots]] of the [[Definition:Quadratic Function|quadratic function]]:
:$\map Q x = a x^2 + b x + c$
whose [[Definition:Discrim... | Because $b^2 - 4 a c > 0$, we have from [[Solution to Quadratic Equation with Real Coefficients]] that the [[Definition:Root of Polynomial|roots]] of $\map Q x$ are [[Definition:Real Number|real]] and unequal.
This demonstrates the existence of $\alpha$ and $\beta$, where [[Definition:By Hypothesis|by hypothesis]] we ... | Sign of Quadratic Function Between Roots | https://proofwiki.org/wiki/Sign_of_Quadratic_Function_Between_Roots | https://proofwiki.org/wiki/Sign_of_Quadratic_Function_Between_Roots | [
"Quadratic Functions"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Root of Polynomial",
"Definition:Quadratic Function",
"Definition:Discriminant of Polynomial/Quadratic Equation",
"Definition:Strictly Positive/Real Number"
] | [
"Solution to Quadratic Equation/Real Coefficients",
"Definition:Root of Polynomial",
"Definition:Real Number",
"Definition:By Hypothesis"
] |
proofwiki-15943 | Minimum Value of Real Quadratic Function | Let $a \in \R_{>0}$ be a (strictly) positive real number.
Consider the quadratic function:
:$\map Q x = a x^2 + b x + c$
$\map Q x$ achieves a minimum at $x = -\dfrac b {2 a}$, at which point $\map Q x = c - \dfrac {b^2} {4 a}$. | {{begin-eqn}}
{{eqn | l = \map Q x
| r = a x^2 + b x + c
| c =
}}
{{eqn | r = \dfrac {4 \paren {a x}^2 + 4 a b x + 4 a c} {4 a}
| c =
}}
{{eqn | r = \dfrac {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } {4 a}
| c =
}}
{{end-eqn}}
As $\paren {2 a x + b}^2 > 0$, it follows that:
{{begin-eqn}}
{... | Let $a \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Consider the [[Definition:Quadratic Function|quadratic function]]:
:$\map Q x = a x^2 + b x + c$
$\map Q x$ achieves a [[Definition:Minimum Value|minimum]] at $x = -\dfrac b {2 a}$, at which point $\map Q x = c - ... | {{begin-eqn}}
{{eqn | l = \map Q x
| r = a x^2 + b x + c
| c =
}}
{{eqn | r = \dfrac {4 \paren {a x}^2 + 4 a b x + 4 a c} {4 a}
| c =
}}
{{eqn | r = \dfrac {\paren {2 a x + b}^2 - \paren {b^2 - 4 a c} } {4 a}
| c =
}}
{{end-eqn}}
As $\paren {2 a x + b}^2 > 0$, it follows that:
{{begin-eqn}... | Minimum Value of Real Quadratic Function | https://proofwiki.org/wiki/Minimum_Value_of_Real_Quadratic_Function | https://proofwiki.org/wiki/Minimum_Value_of_Real_Quadratic_Function | [
"Quadratic Functions"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Quadratic Function",
"Definition:Minimum Value of Real Function/Absolute"
] | [] |
proofwiki-15944 | Number of Type Rational r plus s Root 2 is Irrational | Let $r, s \in \Q$ be rational numbers.
Then $r + s \sqrt 2$ is irrational. | {{AimForCont}} $t = r + s \sqrt 2$ be rational.
Then:
:$\sqrt 2 = \dfrac {t - r} s$ is also rational.
This contradicts the fact that Square Root of 2 is Irrational.
Hence the result by Proof by Contradiction.
{{qed}} | Let $r, s \in \Q$ be [[Definition:Rational Number|rational numbers]].
Then $r + s \sqrt 2$ is [[Definition:Irrational Number|irrational]]. | {{AimForCont}} $t = r + s \sqrt 2$ be [[Definition:Rational Number|rational]].
Then:
:$\sqrt 2 = \dfrac {t - r} s$ is also [[Definition:Rational Number|rational]].
This [[Definition:Contradiction|contradicts]] the fact that [[Square Root of 2 is Irrational]].
Hence the result by [[Proof by Contradiction]].
{{qed}} | Number of Type Rational r plus s Root 2 is Irrational | https://proofwiki.org/wiki/Number_of_Type_Rational_r_plus_s_Root_2_is_Irrational | https://proofwiki.org/wiki/Number_of_Type_Rational_r_plus_s_Root_2_is_Irrational | [
"Real Analysis"
] | [
"Definition:Rational Number",
"Definition:Irrational Number"
] | [
"Definition:Rational Number",
"Definition:Rational Number",
"Definition:Contradiction",
"Square Root of 2 is Irrational",
"Proof by Contradiction"
] |
proofwiki-15945 | Roots of Quadratic with Rational Coefficients of form r plus s Root 2 | Consider the quadratic equation:
:$(1): \quad a^2 x + b x + c = 0$
where $a, b, c$ are rational.
Let $\alpha = r + s \sqrt 2$ be one of the roots of $(1)$.
Then $\beta = r - s \sqrt 2$ is the other root of $(1)$. | We have that:
{{begin-eqn}}
{{eqn | l = a \paren {r + s \sqrt 2}^2 + b \paren {r + s \sqrt 2} + c
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {a r^2 + 2 a s + br + c} + \paren {2 a + b} s \sqrt 2
| r = 0
| c =
}}
{{end-eqn}}
Because $a$, $b$, $c$, $r$ and $s$ are rational, it must... | Consider the [[Definition:Quadratic Equation|quadratic equation]]:
:$(1): \quad a^2 x + b x + c = 0$
where $a, b, c$ are [[Definition:Rational Number|rational]].
Let $\alpha = r + s \sqrt 2$ be one of the [[Definition:Root of Mapping|roots]] of $(1)$.
Then $\beta = r - s \sqrt 2$ is the other [[Definition:Root of Ma... | We have that:
{{begin-eqn}}
{{eqn | l = a \paren {r + s \sqrt 2}^2 + b \paren {r + s \sqrt 2} + c
| r = 0
| c =
}}
{{eqn | ll= \leadsto
| l = \paren {a r^2 + 2 a s + br + c} + \paren {2 a + b} s \sqrt 2
| r = 0
| c =
}}
{{end-eqn}}
Because $a$, $b$, $c$, $r$ and $s$ are [[Definition:Ra... | Roots of Quadratic with Rational Coefficients of form r plus s Root 2 | https://proofwiki.org/wiki/Roots_of_Quadratic_with_Rational_Coefficients_of_form_r_plus_s_Root_2 | https://proofwiki.org/wiki/Roots_of_Quadratic_with_Rational_Coefficients_of_form_r_plus_s_Root_2 | [
"Quadratic Equations"
] | [
"Definition:Quadratic Equation",
"Definition:Rational Number",
"Definition:Root of Mapping",
"Definition:Root of Mapping"
] | [
"Definition:Rational Number",
"Definition:Root of Mapping"
] |
proofwiki-15946 | Descartes's Rule of Signs | Let $\map f x$ be a polynomial equation over the real numbers:
:$a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x + a_0 = 0$
where $a_j \in \R$.
Let $s_n$ be the number of variations in sign of $\map f x$.
Let $p_n$ be the number of positive real roots of $\map f x$ (counted with multiplicity).
Then:
:$\forall n \in \Z... | The proof proceeds by induction.
:For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
:$s_n - p_n = 2 r$ where $r \in \Z_{\ge 0}$ | Let $\map f x$ be a [[Definition:Polynomial Equation|polynomial equation]] over the [[Definition:Real Number|real numbers]]:
:$a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x + a_0 = 0$
where $a_j \in \R$.
Let $s_n$ be the number of [[Definition:Variation in Sign of Polynomial|variations in sign]] of $\map f x$.
Let ... | The proof proceeds by [[Principle of Mathematical Induction|induction]].
:For all $n \in \Z_{>0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$s_n - p_n = 2 r$ where $r \in \Z_{\ge 0}$ | Descartes's Rule of Signs | https://proofwiki.org/wiki/Descartes's_Rule_of_Signs | https://proofwiki.org/wiki/Descartes's_Rule_of_Signs | [
"Descartes's Rule of Signs",
"Polynomial Theory"
] | [
"Definition:Polynomial Equation",
"Definition:Real Number",
"Definition:Variation in Sign of Polynomial",
"Definition:Strictly Positive/Real Number",
"Definition:Real Number",
"Definition:Root of Polynomial",
"Definition:Multiple Root/Multiplicity",
"Definition:Positive/Integer",
"Definition:Even In... | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-15947 | Supremum is not necessarily Greatest Element | Let $\struct {S, \preceq}$ be an ordered set.
Let $T$ admit a supremum in $S$.
Then the supremum of $T$ in $S$ is not necessarily the greatest element of $T$. | Proof by Counterexample:
Consider the subset $T$ of the set of real numbers $\R$:
:$T := \set {x \in \R: 1 \le x < 2}$
The number $2$ cannot be the greatest element of $T$ as $2 \notin T$.
However, $2$ is the supremum of $T$ in $S$.
Indeed, by definition:
:$\forall x \in T: x < 2$
So, let $x < 2$.
Then consider $y = \d... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $T$ admit a [[Definition:Supremum of Set|supremum]] in $S$.
Then the [[Definition:Supremum of Set|supremum]] of $T$ in $S$ is not necessarily the [[Definition:Greatest Element|greatest element]] of $T$. | [[Proof by Counterexample]]:
Consider the [[Definition:Subset|subset]] $T$ of the [[Definition:Real Number|set of real numbers]] $\R$:
:$T := \set {x \in \R: 1 \le x < 2}$
The number $2$ cannot be the [[Definition:Greatest Element|greatest element]] of $T$ as $2 \notin T$.
However, $2$ is the [[Definition:Supremum o... | Supremum is not necessarily Greatest Element/Proof | https://proofwiki.org/wiki/Supremum_is_not_necessarily_Greatest_Element | https://proofwiki.org/wiki/Supremum_is_not_necessarily_Greatest_Element/Proof | [
"Suprema",
"Order Theory",
"Supremum is not necessarily Greatest Element"
] | [
"Definition:Ordered Set",
"Definition:Supremum of Set",
"Definition:Supremum of Set",
"Definition:Greatest Element"
] | [
"Proof by Counterexample",
"Definition:Subset",
"Definition:Real Number",
"Definition:Greatest Element",
"Definition:Supremum of Set",
"Mediant is Between",
"Definition:Greatest Element",
"Definition:Supremum of Set",
"Definition:Greatest Element"
] |
proofwiki-15948 | Infimum is not necessarily Smallest Element | Let $\struct {S, \preceq}$ be an ordered set.
Let $T$ admit a infimum in $S$.
Then the infimum of $T$ in $S$ is not necessarily the smallest element of $T$. | Let $V$ be the subset of the real numbers $\R$ defined as:
:$V := \set {x \in \R: x > 0}$
From Infimum of Subset of Real Numbers: Example 3, $V$ admits an infimum:
:$\inf V = 0$
But $V$ has no smallest element, as follows.
We note that $\inf V = 0 \notin V$.
{{AimForCont}} $x \in V$ is the smallest element of $V$.
Then... | Let $\struct {S, \preceq}$ be an [[Definition:Ordered Set|ordered set]].
Let $T$ admit a [[Definition:Infimum of Set|infimum]] in $S$.
Then the [[Definition:Infimum of Set|infimum]] of $T$ in $S$ is not necessarily the [[Definition:Smallest Element|smallest element]] of $T$. | Let $V$ be the [[Definition:Subset|subset]] of the [[Definition:Real Number|real numbers]] $\R$ defined as:
:$V := \set {x \in \R: x > 0}$
From [[Infimum of Subset of Real Numbers/Examples/Example 3|Infimum of Subset of Real Numbers: Example 3]], $V$ admits an [[Definition:Infimum of Subset of Real Numbers|infimum]]:
... | Infimum is not necessarily Smallest Element/Proof | https://proofwiki.org/wiki/Infimum_is_not_necessarily_Smallest_Element | https://proofwiki.org/wiki/Infimum_is_not_necessarily_Smallest_Element/Proof | [
"Suprema",
"Order Theory",
"Infimum is not necessarily Smallest Element"
] | [
"Definition:Ordered Set",
"Definition:Infimum of Set",
"Definition:Infimum of Set",
"Definition:Smallest Element"
] | [
"Definition:Subset",
"Definition:Real Number",
"Infimum of Subset of Real Numbers/Examples/Example 3",
"Definition:Infimum of Set/Real Numbers",
"Definition:Smallest Element",
"Definition:Smallest Element",
"Definition:Contradiction",
"Definition:Smallest Element",
"Definition:Smallest Element"
] |
proofwiki-15949 | Closed Interval Defined by Absolute Value | :$\set {x \in \R: \size {\xi - x} \le \delta} = \closedint {\xi - \delta} {\xi + \delta}$
where $\closedint {\xi - \delta} {\xi + \delta}$ is the closed real interval between $\xi - \delta$ and $\xi + \delta$. | {{begin-eqn}}
{{eqn | l = \size {\xi - x}
| o = \le
| r = \delta
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -\delta
| o = \le
| r = \xi - x \le \delta
| c = {{Corollary|Negative of Absolute Value|2}}
}}
{{eqn | ll= \leadstoandfrom
| l = \delta
| o = \ge
| r = ... | :$\set {x \in \R: \size {\xi - x} \le \delta} = \closedint {\xi - \delta} {\xi + \delta}$
where $\closedint {\xi - \delta} {\xi + \delta}$ is the [[Definition:Closed Real Interval|closed real interval]] between $\xi - \delta$ and $\xi + \delta$. | {{begin-eqn}}
{{eqn | l = \size {\xi - x}
| o = \le
| r = \delta
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -\delta
| o = \le
| r = \xi - x \le \delta
| c = {{Corollary|Negative of Absolute Value|2}}
}}
{{eqn | ll= \leadstoandfrom
| l = \delta
| o = \ge
| r = ... | Closed Interval Defined by Absolute Value | https://proofwiki.org/wiki/Closed_Interval_Defined_by_Absolute_Value | https://proofwiki.org/wiki/Closed_Interval_Defined_by_Absolute_Value | [
"Real Intervals",
"Absolute Value Function"
] | [
"Definition:Real Interval/Closed"
] | [
"Ordering of Real Numbers is Reversed by Negation",
"Real Number Ordering is Compatible with Addition",
"Definition:Real Interval/Closed"
] |
proofwiki-15950 | Open Interval Defined by Absolute Value | :$\set {x \in \R: \size {\xi - x} < \delta} = \openint {\xi - \delta} {\xi + \delta}$
where $\openint {\xi - \delta} {\xi + \delta}$ is the open real interval between $\xi - \delta$ and $\xi + \delta$. | {{begin-eqn}}
{{eqn | l = \size {\xi - x}
| o = <
| r = \delta
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -\delta
| o = <
| r = \xi - x < \delta
| c = Negative of Absolute Value: Corollary 1
}}
{{eqn | ll= \leadstoandfrom
| l = \delta
| o = >
| r = x - \xi > -... | :$\set {x \in \R: \size {\xi - x} < \delta} = \openint {\xi - \delta} {\xi + \delta}$
where $\openint {\xi - \delta} {\xi + \delta}$ is the [[Definition:Open Real Interval|open real interval]] between $\xi - \delta$ and $\xi + \delta$. | {{begin-eqn}}
{{eqn | l = \size {\xi - x}
| o = <
| r = \delta
| c =
}}
{{eqn | ll= \leadstoandfrom
| l = -\delta
| o = <
| r = \xi - x < \delta
| c = [[Negative of Absolute Value/Corollary 1|Negative of Absolute Value: Corollary 1]]
}}
{{eqn | ll= \leadstoandfrom
| l = \... | Open Interval Defined by Absolute Value | https://proofwiki.org/wiki/Open_Interval_Defined_by_Absolute_Value | https://proofwiki.org/wiki/Open_Interval_Defined_by_Absolute_Value | [
"Real Intervals",
"Absolute Value Function"
] | [
"Definition:Real Interval/Open"
] | [
"Negative of Absolute Value/Corollary 1"
] |
proofwiki-15951 | Set of Strictly Positive Real Numbers has no Smallest Element | Let $\R_{>0}$ denote the set of strictly positive real numbers.
Then $\R_{>0}$ has no smallest element. | {{AimForCont}} $\R_{>0}$ has a smallest element.
Let $m$ be that smallest element.
Then we have that:
:$0 < \dfrac m 2 < m$
But as $0 < \dfrac m 2$ it follows that $\dfrac m 2 \in \R_{>0}$.
This contradicts our assertion that $m$ is the smallest element of $\R_{>0}$.
Hence the result by Proof by Contradiction.
{{qed}} | Let $\R_{>0}$ denote the [[Definition:Strictly Positive Real Number|set of strictly positive real numbers]].
Then $\R_{>0}$ has no [[Definition:Smallest Element|smallest element]]. | {{AimForCont}} $\R_{>0}$ has a [[Definition:Smallest Element|smallest element]].
Let $m$ be that [[Definition:Smallest Element|smallest element]].
Then we have that:
:$0 < \dfrac m 2 < m$
But as $0 < \dfrac m 2$ it follows that $\dfrac m 2 \in \R_{>0}$.
This [[Definition:Contradiction|contradicts]] our assertion th... | Set of Strictly Positive Real Numbers has no Smallest Element | https://proofwiki.org/wiki/Set_of_Strictly_Positive_Real_Numbers_has_no_Smallest_Element | https://proofwiki.org/wiki/Set_of_Strictly_Positive_Real_Numbers_has_no_Smallest_Element | [
"Real Analysis"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Smallest Element"
] | [
"Definition:Smallest Element",
"Definition:Smallest Element",
"Definition:Contradiction",
"Definition:Smallest Element",
"Proof by Contradiction"
] |
proofwiki-15952 | Distance from Subset of Real Numbers to Element | :$x \in S \implies \map d {x, S} = 0$ | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Consider the set $T = \set {\size {x - y}: y \in S}$.
This has $0$ as a lower bound as Absolute Value is Bounded Below by Zero.
So:
:$\ds \map d {x, S} = \m... | :$x \in S \implies \map d {x, S} = 0$ | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Consider the set $T = \set {\size {x - y}: y \in S}$.
This has $0$ as a [[Definition:Lower Bound of Set|low... | Distance from Subset of Real Numbers to Element/Proof 1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Element | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Element/Proof_1 | [
"Distance Function",
"Distance from Subset of Real Numbers to Element"
] | [] | [
"Definition:Distance/Points/Real Numbers",
"Definition:Lower Bound of Set",
"Absolute Value is Bounded Below by Zero"
] |
proofwiki-15953 | Distance from Subset of Real Numbers to Element | :$x \in S \implies \map d {x, S} = 0$ | Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Element.
{{Qed}} | :$x \in S \implies \map d {x, S} = 0$ | Recall from [[Real Number Line is Metric Space]] that the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Distance between Element and Subset of Real Numbers|distance function]] $d$ is a [[Definition:Metric Space|metric space]].
The result is then seen to be an example of [[Distance from Subs... | Distance from Subset of Real Numbers to Element/Proof 2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Element | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Element/Proof_2 | [
"Distance Function",
"Distance from Subset of Real Numbers to Element"
] | [] | [
"Real Number Line is Metric Space",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers",
"Definition:Metric Space",
"Distance from Subset to Element"
] |
proofwiki-15954 | Distance from Subset of Real Numbers to Supremum | Let $S$ be bounded above such that $\xi = \sup S$.
Then:
:$\map d {\xi, S} = 0$ | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \sup S$.
Then:
:$\forall y \in S: \size {\xi - y} = \xi - y$
So we need to show that no $h > 0$ can be a lower bound for $T = \set {\size {\xi - ... | Let $S$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]] such that $\xi = \sup S$.
Then:
:$\map d {\xi, S} = 0$ | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \sup S$.
Then:
:$\forall y \in S: \size {\xi - y} = \xi - y$
So we need to show that no $h > 0$... | Distance from Subset of Real Numbers to Supremum/Proof 1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Supremum | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Supremum/Proof_1 | [
"Distance Function",
"Distance from Subset of Real Numbers to Supremum"
] | [
"Definition:Bounded Above Set/Real Numbers"
] | [
"Definition:Distance/Points/Real Numbers",
"Definition:Lower Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Contradiction",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Proof by Contradiction"
] |
proofwiki-15955 | Distance from Subset of Real Numbers to Supremum | Let $S$ be bounded above such that $\xi = \sup S$.
Then:
:$\map d {\xi, S} = 0$ | Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Supremum.
{{Qed}} | Let $S$ be [[Definition:Bounded Above Subset of Real Numbers|bounded above]] such that $\xi = \sup S$.
Then:
:$\map d {\xi, S} = 0$ | Recall from [[Real Number Line is Metric Space]] that the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Distance between Element and Subset of Real Numbers|distance function]] $d$ is a [[Definition:Metric Space|metric space]].
The result is then seen to be an example of [[Distance from Subs... | Distance from Subset of Real Numbers to Supremum/Proof 2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Supremum | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Supremum/Proof_2 | [
"Distance Function",
"Distance from Subset of Real Numbers to Supremum"
] | [
"Definition:Bounded Above Set/Real Numbers"
] | [
"Real Number Line is Metric Space",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers",
"Definition:Metric Space",
"Distance from Subset to Supremum"
] |
proofwiki-15956 | Distance from Subset of Real Numbers to Infimum | Let $S$ be bounded below such that $\xi = \inf S$.
Then:
:$\map d {\xi, S} = 0$ | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \inf S$.
Consider $\map d {-\xi, S'}$ where $S' = \set {-\xi: \xi \in S}$.
By Negative of Infimum is Supremum of Negatives:
:$\xi = \inf S \impli... | Let $S$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]] such that $\xi = \inf S$.
Then:
:$\map d {\xi, S} = 0$ | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \inf S$.
Consider $\map d {-\xi, S'}$ where $S' = \set {-\xi: \xi \in S}$.
By [[Negative of Inf... | Distance from Subset of Real Numbers to Infimum/Proof 1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Infimum | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Infimum/Proof_1 | [
"Distance Function",
"Distance from Subset of Real Numbers to Infimum"
] | [
"Definition:Bounded Below Set/Real Numbers"
] | [
"Definition:Distance/Points/Real Numbers",
"Negative of Infimum is Supremum of Negatives"
] |
proofwiki-15957 | Distance from Subset of Real Numbers to Infimum | Let $S$ be bounded below such that $\xi = \inf S$.
Then:
:$\map d {\xi, S} = 0$ | Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Infimum.
{{Qed}} | Let $S$ be [[Definition:Bounded Below Subset of Real Numbers|bounded below]] such that $\xi = \inf S$.
Then:
:$\map d {\xi, S} = 0$ | Recall from [[Real Number Line is Metric Space]] that the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Distance between Element and Subset of Real Numbers|distance function]] $d$ is a [[Definition:Metric Space|metric space]].
The result is then seen to be an example of [[Distance from Subs... | Distance from Subset of Real Numbers to Infimum/Proof 2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Infimum | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Infimum/Proof_2 | [
"Distance Function",
"Distance from Subset of Real Numbers to Infimum"
] | [
"Definition:Bounded Below Set/Real Numbers"
] | [
"Real Number Line is Metric Space",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers",
"Definition:Metric Space",
"Distance from Subset to Infimum"
] |
proofwiki-15958 | Real Number at Distance Zero from Closed Real Interval is In Interval | Let $I \subseteq \R$ be a closed real interval.
Then:
:$\map d {x, I} = 0 \implies x \in I$ | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Because $I$ is an interval, if $x \notin I$ then $x$ is either an upper bound or a lower bound for $I$.
Suppose $x$ is an upper bound for $I$.
Let $B$ be th... | Let $I \subseteq \R$ be a [[Definition:Closed Real Interval|closed real interval]].
Then:
:$\map d {x, I} = 0 \implies x \in I$ | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Because $I$ is an [[Definition:Real Interval|interval]], if $x \notin I$ then $x$ is either an [[Definition:... | Real Number at Distance Zero from Closed Real Interval is In Interval | https://proofwiki.org/wiki/Real_Number_at_Distance_Zero_from_Closed_Real_Interval_is_In_Interval | https://proofwiki.org/wiki/Real_Number_at_Distance_Zero_from_Closed_Real_Interval_is_In_Interval | [
"Distance Function"
] | [
"Definition:Real Interval/Closed"
] | [
"Definition:Distance/Points/Real Numbers",
"Definition:Real Interval",
"Definition:Upper Bound of Set",
"Definition:Lower Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Supremum of Set",
"Definition:Real Interval/Closed",
"Infimum Plus Constant",
"Definition:Lower Bound of Set"
] |
proofwiki-15959 | Existence of Real Number at Distance Zero from Open Real Interval not in Interval | Let $I \subseteq \R$ be an open real interval such that $I \ne \O$ and $I \ne \R$.
Then:
:$\exists x \notin I: \map d {x, I} = 0$ | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
As $I \ne \O$ and $I \ne \R$ it follows that one of the following applies:
{{begin-eqn}}
{{eqn | q = \exists a, b \in \R
| l = I
| r = \openint ... | Let $I \subseteq \R$ be an [[Definition:Open Real Interval|open real interval]] such that $I \ne \O$ and $I \ne \R$.
Then:
:$\exists x \notin I: \map d {x, I} = 0$ | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
As $I \ne \O$ and $I \ne \R$ it follows that one of the following applies:
{{begin-eqn}}
{{eqn | q = \exist... | Existence of Real Number at Distance Zero from Open Real Interval not in Interval | https://proofwiki.org/wiki/Existence_of_Real_Number_at_Distance_Zero_from_Open_Real_Interval_not_in_Interval | https://proofwiki.org/wiki/Existence_of_Real_Number_at_Distance_Zero_from_Open_Real_Interval_not_in_Interval | [
"Distance Function"
] | [
"Definition:Real Interval/Open"
] | [
"Definition:Distance/Points/Real Numbers",
"Definition:Real Interval/Open",
"Definition:Infimum of Set",
"Definition:Supremum of Set"
] |
proofwiki-15960 | Distance from Subset of Real Numbers to Element/Proof 1 | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Then:
{{:Distance from Subset of Real Numbers to Element}} | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Consider the set $T = \set {\size {x - y}: y \in S}$.
This has $0$ as a lower bound as Absolute Value is Bounded Below by Zero.
So:
:$\ds \map d {x, S} = \m... | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
Then:
{{:Distance from Subset ... | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Consider the set $T = \set {\size {x - y}: y \in S}$.
This has $0$ as a [[Definition:Lower Bound of Set|low... | Distance from Subset of Real Numbers to Element/Proof 1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Element/Proof_1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Element/Proof_1 | [
"Distance from Subset of Real Numbers to Element"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Definition:Distance/Points/Real Numbers",
"Definition:Lower Bound of Set",
"Absolute Value is Bounded Below by Zero"
] |
proofwiki-15961 | Distance from Subset of Real Numbers to Element/Proof 2 | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
Then:
{{:Distance from Subset of Real Numbers to Element}} | Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Element.
{{Qed}} | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
Then:
{{:Distance from Subset ... | Recall from [[Real Number Line is Metric Space]] that the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Distance between Element and Subset of Real Numbers|distance function]] $d$ is a [[Definition:Metric Space|metric space]].
The result is then seen to be an example of [[Distance from Subs... | Distance from Subset of Real Numbers to Element/Proof 2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Element/Proof_2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Element/Proof_2 | [
"Distance from Subset of Real Numbers to Element"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Real Number Line is Metric Space",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers",
"Definition:Metric Space",
"Distance from Subset to Element"
] |
proofwiki-15962 | Distance from Subset of Real Numbers to Supremum/Proof 1 | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
{{:Distance from Subset of Real Numbers to Supremum}} | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \sup S$.
Then:
:$\forall y \in S: \size {\xi - y} = \xi - y$
So we need to show that no $h > 0$ can be a lower bound for $T = \set {\size {\xi - ... | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
{{:Distance from Subset of Real... | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \sup S$.
Then:
:$\forall y \in S: \size {\xi - y} = \xi - y$
So we need to show that no $h > 0$... | Distance from Subset of Real Numbers to Supremum/Proof 1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Supremum/Proof_1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Supremum/Proof_1 | [
"Distance from Subset of Real Numbers to Supremum"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Definition:Distance/Points/Real Numbers",
"Definition:Lower Bound of Set",
"Definition:Upper Bound of Set",
"Definition:Contradiction",
"Definition:Supremum of Set",
"Definition:Upper Bound of Set",
"Proof by Contradiction"
] |
proofwiki-15963 | Distance from Subset of Real Numbers to Supremum/Proof 2 | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
{{:Distance from Subset of Real Numbers to Supremum}} | Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Supremum.
{{Qed}} | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
{{:Distance from Subset of Real... | Recall from [[Real Number Line is Metric Space]] that the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Distance between Element and Subset of Real Numbers|distance function]] $d$ is a [[Definition:Metric Space|metric space]].
The result is then seen to be an example of [[Distance from Subs... | Distance from Subset of Real Numbers to Supremum/Proof 2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Supremum/Proof_2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Supremum/Proof_2 | [
"Distance from Subset of Real Numbers to Supremum"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Real Number Line is Metric Space",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers",
"Definition:Metric Space",
"Distance from Subset to Supremum"
] |
proofwiki-15964 | Distance from Subset of Real Numbers to Infimum/Proof 1 | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
{{:Distance from Subset of Real Numbers to Infimum}} | From the definition of distance:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \inf S$.
Consider $\map d {-\xi, S'}$ where $S' = \set {-\xi: \xi \in S}$.
By Negative of Infimum is Supremum of Negatives:
:$\xi = \inf S \impli... | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
{{:Distance from Subset of Real... | From the definition of [[Definition:Distance between Real Numbers|distance]]:
:$\forall x, y \in \R: \map d {x, y} = \size {x - y}$
Thus:
:$\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\size {x - y} }$
Let $\xi = \inf S$.
Consider $\map d {-\xi, S'}$ where $S' = \set {-\xi: \xi \in S}$.
By [[Negative of Inf... | Distance from Subset of Real Numbers to Infimum/Proof 1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Infimum/Proof_1 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Infimum/Proof_1 | [
"Distance from Subset of Real Numbers to Infimum"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Definition:Distance/Points/Real Numbers",
"Negative of Infimum is Supremum of Negatives"
] |
proofwiki-15965 | Distance from Subset of Real Numbers to Infimum/Proof 2 | Let $S$ be a subset of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
Let $\map d {x, S}$ be the distance between $x$ and $S$.
{{:Distance from Subset of Real Numbers to Infimum}} | Recall from Real Number Line is Metric Space that the set of real numbers $\R$ with the distance function $d$ is a metric space.
The result is then seen to be an example of Distance from Subset to Infimum.
{{Qed}} | Let $S$ be a [[Definition:Subset|subset]] of the set of [[Definition:Real Number|real numbers]] $\R$.
Let $x \in \R$ be a [[Definition:Real Number|real number]].
Let $\map d {x, S}$ be the [[Definition:Distance between Element and Subset of Real Numbers|distance]] between $x$ and $S$.
{{:Distance from Subset of Real... | Recall from [[Real Number Line is Metric Space]] that the [[Definition:Real Number|set of real numbers]] $\R$ with the [[Definition:Distance between Element and Subset of Real Numbers|distance function]] $d$ is a [[Definition:Metric Space|metric space]].
The result is then seen to be an example of [[Distance from Subs... | Distance from Subset of Real Numbers to Infimum/Proof 2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Infimum/Proof_2 | https://proofwiki.org/wiki/Distance_from_Subset_of_Real_Numbers_to_Infimum/Proof_2 | [
"Distance from Subset of Real Numbers to Infimum"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers"
] | [
"Real Number Line is Metric Space",
"Definition:Real Number",
"Definition:Distance/Sets/Real Numbers",
"Definition:Metric Space",
"Distance from Subset to Infimum"
] |
proofwiki-15966 | Infimum of Set of Reciprocals of Positive Integers | Let $S$ be the subset of the set of real numbers defined as:
:$S = \set {\dfrac 1 n: n \in \Z_{>0} }$
Then:
:$\inf S = 0$
where $\inf S$ denotes the infimum of $S$. | We have that $\Z_{>0}$ contains only (strictly) positive integers.
So it follows from Reciprocal of Strictly Positive Real Number is Strictly Positive that $S$ contains only (strictly) positive real numbers.
Hence $0$ is a lower bound for $S$.
{{AimForCont}} that $0$ is not the infimum of $S$.
Then $\exists h \in \R_{>... | Let $S$ be the [[Definition:Subset|subset]] of the [[Definition:Real Number|set of real numbers]] defined as:
:$S = \set {\dfrac 1 n: n \in \Z_{>0} }$
Then:
:$\inf S = 0$
where $\inf S$ denotes the [[Definition:Infimum of Subset of Real Numbers|infimum]] of $S$. | We have that $\Z_{>0}$ contains only [[Definition:Strictly Positive Integer|(strictly) positive integers]].
So it follows from [[Reciprocal of Strictly Positive Real Number is Strictly Positive]] that $S$ contains only [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]].
Hence $0$ is a [[Def... | Infimum of Set of Reciprocals of Positive Integers | https://proofwiki.org/wiki/Infimum_of_Set_of_Reciprocals_of_Positive_Integers | https://proofwiki.org/wiki/Infimum_of_Set_of_Reciprocals_of_Positive_Integers | [
"Reciprocals",
"Infima"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Infimum of Set/Real Numbers"
] | [
"Definition:Strictly Positive/Integer",
"Reciprocal of Strictly Positive Real Number is Strictly Positive",
"Definition:Strictly Positive/Real Number",
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers",
"Definition:Lower Bound of Set/Real Numbers",
"Reciprocal Functi... |
proofwiki-15967 | Set of Numbers of form n - 1 over n is Bounded Above | Let $S$ be the subset of the set of real numbers $\R$ defined as:
:$S = \set {\dfrac {n - 1} n: n \in \Z_{>0} }$
$S$ is bounded above with supremum $1$.
$S$ has no greatest element. | We have that:
:$\dfrac {n - 1} n = 1 - \dfrac 1 n$
As $n > 0$ it follows from Reciprocal of Strictly Positive Real Number is Strictly Positive that $\dfrac 1 n > 0$.
Thus $1 - \dfrac 1 n < 1$ and so $S$ is bounded above by $1$.
Next it is to be shown that $1$ is the supremum of $S$.
Suppose $x$ is the supremum of $S$ s... | Let $S$ be the [[Definition:Subset|subset]] of the [[Definition:Real Numbers|set of real numbers]] $\R$ defined as:
:$S = \set {\dfrac {n - 1} n: n \in \Z_{>0} }$
$S$ is [[Definition:Bounded Above Subset of Real Numbers|bounded above]] with [[Definition:Supremum of Subset of Real Numbers|supremum]] $1$.
$S$ has no ... | We have that:
:$\dfrac {n - 1} n = 1 - \dfrac 1 n$
As $n > 0$ it follows from [[Reciprocal of Strictly Positive Real Number is Strictly Positive]] that $\dfrac 1 n > 0$.
Thus $1 - \dfrac 1 n < 1$ and so $S$ is [[Definition:Bounded Above Subset of Real Numbers|bounded above]] by $1$.
Next it is to be shown that $1$... | Set of Numbers of form n - 1 over n is Bounded Above | https://proofwiki.org/wiki/Set_of_Numbers_of_form_n_-_1_over_n_is_Bounded_Above | https://proofwiki.org/wiki/Set_of_Numbers_of_form_n_-_1_over_n_is_Bounded_Above | [
"Suprema"
] | [
"Definition:Subset",
"Definition:Real Number",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Definition:Greatest Element"
] | [
"Reciprocal of Strictly Positive Real Number is Strictly Positive",
"Definition:Bounded Above Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Definition:Supremum of Set/Real Numbers",
"Axiom of Archimedes",
"Reciprocal Function is Strictly Decreasing",
"Definition:Supremum of Set/Real Num... |
proofwiki-15968 | Set of Rational Numbers Strictly between Zero and One has no Greatest or Least Element | Let $S \subseteq \Q$ be the subset of the set of rational numbers defined as:
:$S = \set {r \in \Q: 0 < r < 1}$
Then $S$ has no greatest or smallest element.
However, $S$ has a supremum $1$ and an infimum $0$. | We have that:
:$\forall r \in S: 0 < r$
and:
:$\forall r \in S: r < 1$
Hence $0$ and $1$ are lower and upper bounds of $S$ respectively.
Let $s \in S$.
Then $s \in \Q: 0 < s < 1$.
{{AimForCont}} $s$ is the greatest element of $S$.
But then we have:
:$0 < s < \dfrac {s + 1} 2 < 1$
and so $\dfrac {s + 1} 2 \in S$ but $s ... | Let $S \subseteq \Q$ be the [[Definition:Subset|subset]] of the [[Definition:Rational Number|set of rational numbers]] defined as:
:$S = \set {r \in \Q: 0 < r < 1}$
Then $S$ has no [[Definition:Greatest Element|greatest]] or [[Definition:Smallest Element|smallest element]].
However, $S$ has a [[Definition:Supremum ... | We have that:
:$\forall r \in S: 0 < r$
and:
:$\forall r \in S: r < 1$
Hence $0$ and $1$ are [[Definition:Lower Bound of Subset of Real Numbers|lower]] and [[Definition:Upper Bound of Subset of Real Numbers|upper bounds]] of $S$ respectively.
Let $s \in S$.
Then $s \in \Q: 0 < s < 1$.
{{AimForCont}} $s$ is the [[D... | Set of Rational Numbers Strictly between Zero and One has no Greatest or Least Element | https://proofwiki.org/wiki/Set_of_Rational_Numbers_Strictly_between_Zero_and_One_has_no_Greatest_or_Least_Element | https://proofwiki.org/wiki/Set_of_Rational_Numbers_Strictly_between_Zero_and_One_has_no_Greatest_or_Least_Element | [
"Rational Numbers",
"Infima",
"Suprema"
] | [
"Definition:Subset",
"Definition:Rational Number",
"Definition:Greatest Element",
"Definition:Smallest Element",
"Definition:Supremum of Set/Real Numbers",
"Definition:Infimum of Set/Real Numbers"
] | [
"Definition:Lower Bound of Set/Real Numbers",
"Definition:Upper Bound of Set/Real Numbers",
"Definition:Greatest Element",
"Definition:Contradiction",
"Definition:Greatest Element",
"Definition:Smallest Element",
"Definition:Contradiction",
"Definition:Smallest Element",
"Definition:Greatest Element... |
proofwiki-15969 | Between two Real Numbers exists Irrational Number | Let $a, b \in \R$ be real numbers where $a < b$.
Then there exists an irrational number $\xi \in \R \setminus \Q$ such that:
:$a < \xi < b$ | From Number of Type Rational r plus s Root 2 is Irrational we have that a real number of the form $r \sqrt 2$, where $r \ne 0$ is rational, is irrational.
From Between two Real Numbers exists Rational Number there exists a rational number $r$ such that:
:$\dfrac a {\sqrt 2} < r < \dfrac b {\sqrt 2}$
and so:
:$a < r \sq... | Let $a, b \in \R$ be [[Definition:Real Number|real numbers]] where $a < b$.
Then there exists an [[Definition:Irrational Number|irrational number]] $\xi \in \R \setminus \Q$ such that:
:$a < \xi < b$ | From [[Number of Type Rational r plus s Root 2 is Irrational]] we have that a [[Definition:Real Number|real number]] of the form $r \sqrt 2$, where $r \ne 0$ is [[Definition:Rational Number|rational]], is [[Definition:Irrational Number|irrational]].
From [[Between two Real Numbers exists Rational Number]] there exists... | Between two Real Numbers exists Irrational Number | https://proofwiki.org/wiki/Between_two_Real_Numbers_exists_Irrational_Number | https://proofwiki.org/wiki/Between_two_Real_Numbers_exists_Irrational_Number | [
"Real Analysis"
] | [
"Definition:Real Number",
"Definition:Irrational Number"
] | [
"Number of Type Rational r plus s Root 2 is Irrational",
"Definition:Real Number",
"Definition:Rational Number",
"Definition:Irrational Number",
"Between two Real Numbers exists Rational Number",
"Definition:Rational Number",
"Definition:Irrational Number",
"Definition:Zero (Number)",
"Definition:Ra... |
proofwiki-15970 | Difference of Two Powers/General Commutative Ring | Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$.
Let $a, b \in R$.
Let $n \in \N$ such that $n \ge 2$.
Then:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \circ \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} \circ b^j
| c =
}}
{{eqn | r = \paren {a - b} \circ \paren {a^{n - 1} ... | Let $\ds S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} \circ b^j$.
This can also be written:
:$\ds S_n = \sum_{j \mathop = 0}^{n - 1} b^j \circ a^{n - j - 1}$
Consider:
:$\ds a \circ S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j} \circ b^j$
Taking the first term (where $j = 0$) out of the summation, we get:
:$\ds a \c... | Let $\struct {R, +, \circ}$ be a [[Definition:Commutative Ring|commutative ring]] whose [[Definition:Ring Zero|zero]] is $0_R$.
Let $a, b \in R$.
Let $n \in \N$ such that $n \ge 2$.
Then:
{{begin-eqn}}
{{eqn | l = a^n - b^n
| r = \paren {a - b} \circ \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} \circ b^j
... | Let $\ds S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j - 1} \circ b^j$.
This can also be written:
:$\ds S_n = \sum_{j \mathop = 0}^{n - 1} b^j \circ a^{n - j - 1}$
Consider:
:$\ds a \circ S_n = \sum_{j \mathop = 0}^{n - 1} a^{n - j} \circ b^j$
Taking the first term (where $j = 0$) out of the summation, we get:
:$\ds ... | Difference of Two Powers/General Commutative Ring | https://proofwiki.org/wiki/Difference_of_Two_Powers/General_Commutative_Ring | https://proofwiki.org/wiki/Difference_of_Two_Powers/General_Commutative_Ring | [
"Commutative Rings",
"Polynomial Theory",
"Difference of Two Powers"
] | [
"Definition:Commutative Ring",
"Definition:Ring Zero"
] | [
"Permutation of Indices of Summation",
"Category:Commutative Rings",
"Category:Polynomial Theory",
"Category:Difference of Two Powers"
] |
proofwiki-15971 | Sequence of Powers of Reciprocals is Null Sequence/Real Index | Let $r \in \R_{>0}$ be a strictly positive real number.
Let $\sequence {x_n}$ be the sequence in $\R$ defined as:
: $x_n = \dfrac 1 {n^r}$
Then $\sequence {x_n}$ is a null sequence. | Let $\epsilon > 0$.
We need to show that:
:$\exists N \in \N: n > N \implies \size {\dfrac 1 {n^r} } < \epsilon$
That is, that $n^r > 1 / \epsilon$.
Let us choose $N = \ceiling {\paren {1 / \epsilon}^{1/r} }$.
By Reciprocal of Strictly Positive Real Number is Strictly Positive and power of positive real number is posit... | Let $r \in \R_{>0}$ be a [[Definition:Strictly Positive Real Number|strictly positive real number]].
Let $\sequence {x_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as:
: $x_n = \dfrac 1 {n^r}$
Then $\sequence {x_n}$ is a [[Definition:Null Sequence (Analysis)|null sequence]]. | Let $\epsilon > 0$.
We need to show that:
:$\exists N \in \N: n > N \implies \size {\dfrac 1 {n^r} } < \epsilon$
That is, that $n^r > 1 / \epsilon$.
Let us choose $N = \ceiling {\paren {1 / \epsilon}^{1/r} }$.
By [[Reciprocal of Strictly Positive Real Number is Strictly Positive]] and [[Power of Positive Real Numb... | Sequence of Powers of Reciprocals is Null Sequence/Real Index | https://proofwiki.org/wiki/Sequence_of_Powers_of_Reciprocals_is_Null_Sequence/Real_Index | https://proofwiki.org/wiki/Sequence_of_Powers_of_Reciprocals_is_Null_Sequence/Real_Index | [
"Sequence of Powers of Reciprocals is Null Sequence"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Real Sequence",
"Definition:Null Sequence/Analysis"
] | [
"Reciprocal of Strictly Positive Real Number is Strictly Positive",
"Power of Positive Real Number is Positive/Rational Number",
"Positive Power Function on Non-negative Reals is Strictly Increasing"
] |
proofwiki-15972 | Odd Order Derivative of Even Function Vanishes at Zero | Let $X$ be a symmetric subset of $\R$ containing $0$.
Let $n$ be a positive integer.
Let $f:X \to \R$ be an even function.
Let $f$ be at least $\paren{2 n + 1}$-times differentiable.
Then:
:$\map {f^{\paren {2 n + 1} } } 0 = 0$ | From the definition of an even function, for all $x \in X$ we have:
:$\map f x = \map f {-x}$
Differentiating $2 n + 1$ times, we have, by the Chain Rule for Derivatives:
:$\map {f^{\paren {2 n + 1} } } x = \paren {-1}^{2 n + 1} \map {f^{\paren {2 n + 1} } } {-x} = -\map {f^{\paren {2 n + 1} } } {-x}$
Setting $x = 0... | Let $X$ be a [[Definition:Symmetric Set of Real Numbers|symmetric subset]] of $\R$ containing $0$.
Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $f:X \to \R$ be an [[Definition:Even Function|even function]].
Let $f$ be at least $\paren{2 n + 1}$-times [[Definition:Differentiable Function|diff... | From the definition of an [[Definition:Even Function|even function]], for all $x \in X$ we have:
:$\map f x = \map f {-x}$
[[Definition:Differentiation|Differentiating]] $2 n + 1$ times, we have, by the [[Chain Rule for Derivatives]]:
:$\map {f^{\paren {2 n + 1} } } x = \paren {-1}^{2 n + 1} \map {f^{\paren {2 n ... | Odd Order Derivative of Even Function Vanishes at Zero | https://proofwiki.org/wiki/Odd_Order_Derivative_of_Even_Function_Vanishes_at_Zero | https://proofwiki.org/wiki/Odd_Order_Derivative_of_Even_Function_Vanishes_at_Zero | [
"Differential Calculus",
"Even Functions"
] | [
"Definition:Symmetric Set/Real Numbers",
"Definition:Positive/Integer",
"Definition:Even Function",
"Definition:Differentiable Mapping"
] | [
"Definition:Even Function",
"Definition:Differentiation",
"Derivative of Composite Function",
"Category:Differential Calculus",
"Category:Even Functions"
] |
proofwiki-15973 | Reciprocal of Null Sequence/Corollary | :$x_n \to \infty$ as $n \to \infty$ {{iff}} $\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$ | Let $\sequence {y_n}$ be the sequence in $\R$ defined as:
:$y_n = \size {\dfrac 1 {x_n} }$
From Reciprocal of Null Sequence:
:$y_n \to 0$ as $n \to \infty$ {{iff}} $\size {\dfrac 1 {y_n} } \to \infty$ as $n \to \infty$
That is:
:$\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$ {{iff}} $x_n \to \infty$ as $n \to \infty... | :$x_n \to \infty$ as $n \to \infty$ {{iff}} $\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$ | Let $\sequence {y_n}$ be the [[Definition:Real Sequence|sequence in $\R$]] defined as:
:$y_n = \size {\dfrac 1 {x_n} }$
From [[Reciprocal of Null Sequence]]:
:$y_n \to 0$ as $n \to \infty$ {{iff}} $\size {\dfrac 1 {y_n} } \to \infty$ as $n \to \infty$
That is:
:$\size {\dfrac 1 {x_n} } \to 0$ as $n \to \infty$ {{... | Reciprocal of Null Sequence/Corollary | https://proofwiki.org/wiki/Reciprocal_of_Null_Sequence/Corollary | https://proofwiki.org/wiki/Reciprocal_of_Null_Sequence/Corollary | [
"Reciprocal of Null Sequence"
] | [] | [
"Definition:Real Sequence",
"Reciprocal of Null Sequence"
] |
proofwiki-15974 | Unbounded Monotone Sequence Diverges to Infinity/Increasing | Let $\sequence {x_n}$ be increasing and unbounded above.
Then $x_n \to +\infty$ as $n \to \infty$. | Let $H > 0$.
As $\sequence {x_n}$ is unbounded above:
:$\exists N: x_N > H$
As $\sequence {x_n}$ is increasing:
:$\forall n \ge N: x_n \ge x_N > H$
It follows from the definition of divergence to $+\infty$ that $x_n \to +\infty$ as $n \to \infty$.
{{qed}} | Let $\sequence {x_n}$ be [[Definition:Increasing Real Sequence|increasing]] and [[Definition:Unbounded Above Real Sequence|unbounded above]].
Then $x_n \to +\infty$ as $n \to \infty$. | Let $H > 0$.
As $\sequence {x_n}$ is [[Definition:Unbounded Above Real Sequence|unbounded above]]:
:$\exists N: x_N > H$
As $\sequence {x_n}$ is [[Definition:Increasing Real Sequence|increasing]]:
:$\forall n \ge N: x_n \ge x_N > H$
It follows from the definition of [[Definition:Divergent Real Sequence to Positive I... | Unbounded Monotone Sequence Diverges to Infinity/Increasing | https://proofwiki.org/wiki/Unbounded_Monotone_Sequence_Diverges_to_Infinity/Increasing | https://proofwiki.org/wiki/Unbounded_Monotone_Sequence_Diverges_to_Infinity/Increasing | [
"Unbounded Monotone Sequence Diverges to Infinity"
] | [
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Bounded Above Sequence/Real/Unbounded"
] | [
"Definition:Bounded Above Sequence/Real/Unbounded",
"Definition:Increasing/Sequence/Real Sequence",
"Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity"
] |
proofwiki-15975 | Index of Subsequence not Less than its Index | Let $\sequence {x_n}_{n \mathop \ge 1}$ be a sequence in a set $S$.
Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$.
Then:
:$\forall n \in \N_{>0}: n_r \ge r$ | The proof proceeds by induction.
For all $r \in \Z_{\ge 1}$, let $\map P r$ be the proposition:
:$n_r \ge r$ | Let $\sequence {x_n}_{n \mathop \ge 1}$ be a [[Definition:Sequence|sequence]] in a [[Definition:Set|set]] $S$.
Let $\sequence {x_{n_r} }$ be a [[Definition:Subsequence|subsequence]] of $\sequence {x_n}$.
Then:
:$\forall n \in \N_{>0}: n_r \ge r$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $r \in \Z_{\ge 1}$, let $\map P r$ be the [[Definition:Proposition|proposition]]:
:$n_r \ge r$ | Index of Subsequence not Less than its Index | https://proofwiki.org/wiki/Index_of_Subsequence_not_Less_than_its_Index | https://proofwiki.org/wiki/Index_of_Subsequence_not_Less_than_its_Index | [
"Subsequences"
] | [
"Definition:Sequence",
"Definition:Set",
"Definition:Subsequence"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-15976 | Propagation of Light in Inhomogeneous Medium | Let $v: \R^3 \to \R$ be a real function.
Let $M$ be a 3-dimensional Euclidean space.
Let $\gamma:t \in \R \to M$ be a smooth curve embedded in $M$, where $t$ is time.
Denote its derivative {{WRT}} time by $v$.
Suppose $M$ is filled with an optically inhomogeneous medium such that at each point speed of light is $v = \m... | By assumption, $\map y x$ and $\map z x$ are real functions.
This allows us to $x$ instead of $t$ to parameterize the curve.
This reduces the number of equations of motion to $2$, that is: $\map y x$ and $\map z x$.
The time it takes to traverse the curve $\gamma$ equals:
{{begin-eqn}}
{{eqn | l = T
| r = \int_{t... | Let $v: \R^3 \to \R$ be a [[Definition:Real Function|real function]].
Let $M$ be a 3-[[Definition:Dimension of Vector Space|dimensional]] [[Definition:Euclidean Space|Euclidean space]].
Let $\gamma:t \in \R \to M$ be a [[Definition:Smooth Curve|smooth curve]] embedded in $M$, where $t$ is [[Definition:Time|time]].
D... | By [[Definition:Assumption|assumption]], $\map y x$ and $\map z x$ are [[Definition:Real Function|real functions]].
This allows us to $x$ instead of $t$ to [[Definition:Parameterization of Directed Smooth Curve|parameterize]] the [[Definition:Curve|curve]].
This reduces the number of equations of motion to $2$, that ... | Propagation of Light in Inhomogeneous Medium | https://proofwiki.org/wiki/Propagation_of_Light_in_Inhomogeneous_Medium | https://proofwiki.org/wiki/Propagation_of_Light_in_Inhomogeneous_Medium | [
"Calculus of Variations"
] | [
"Definition:Real Function",
"Definition:Dimension of Vector Space",
"Definition:Euclidean Space",
"Definition:Smooth Curve",
"Definition:Time",
"Definition:Derivative",
"Definition:Time",
"Definition:Point",
"Definition:Speed of Light",
"Definition:Real Function",
"Definition:Light (Radiation)",... | [
"Definition:Assumption",
"Definition:Real Function",
"Definition:Directed Smooth Curve/Parameterization",
"Definition:Line/Curve",
"Definition:Time",
"Definition:Line/Curve",
"Derivative of Composite Function",
"Definition:Arc Length",
"Derivative of Composite Function",
"Definition:Dimension of V... |
proofwiki-15977 | Even Order Derivative of Odd Function Vanishes at Zero | Let $X$ be a symmetric subset of $\R$ containing $0$.
Let $n$ be a positive integer.
Let $f:X \to \R$ be an odd function.
Let $f$ be at least $\paren {2 n}$-times differentiable.
Then:
:$\map {f^{\paren {2 n} } } 0 = 0$ | From the definition of an odd function, for all $x \in X$ we have:
:$\map f x = -\map f {-x}$
Differentiating $2 n$ times, we have, by the Chain Rule for Derivatives:
:$\map {f^{\paren {2 n} } } x = -\paren {-1}^{2 n} \map {f^{\paren {2 n} } } {-x} = -\map {f^{\paren {2 n} } } {-x}$
Setting $x = 0$ gives:
:$\map {f^{\... | Let $X$ be a [[Definition:Symmetric Set of Real Numbers|symmetric subset]] of $\R$ containing $0$.
Let $n$ be a [[Definition:Positive Integer|positive integer]].
Let $f:X \to \R$ be an [[Definition:Odd Function|odd function]].
Let $f$ be at least $\paren {2 n}$-times [[Definition:Differentiable Function|different... | From the definition of an odd function, for all $x \in X$ we have:
:$\map f x = -\map f {-x}$
Differentiating $2 n$ times, we have, by the [[Chain Rule for Derivatives]]:
:$\map {f^{\paren {2 n} } } x = -\paren {-1}^{2 n} \map {f^{\paren {2 n} } } {-x} = -\map {f^{\paren {2 n} } } {-x}$
Setting $x = 0$ gives:
:$\... | Even Order Derivative of Odd Function Vanishes at Zero | https://proofwiki.org/wiki/Even_Order_Derivative_of_Odd_Function_Vanishes_at_Zero | https://proofwiki.org/wiki/Even_Order_Derivative_of_Odd_Function_Vanishes_at_Zero | [
"Differential Calculus",
"Odd Functions"
] | [
"Definition:Symmetric Set/Real Numbers",
"Definition:Positive/Integer",
"Definition:Odd Function",
"Definition:Differentiable Mapping"
] | [
"Derivative of Composite Function",
"Category:Differential Calculus",
"Category:Odd Functions"
] |
proofwiki-15978 | Geometric Mean of two Positive Real Numbers is Between them | Let $a, b \in \R$ be real numbers such that $0 < a < b$.
Let $\map G {a, b}$ denote the geometric mean of $a$ and $b$.
Then:
:$a < \map G {a, b} < b$ | By definition of geometric mean:
:$\map G {a, b} := \sqrt {a b}$
where $\sqrt {a b}$ specifically denotes the positive square root of $a$ and $b$.
Thus:
{{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c =
}}
{{eqn | ll= \leadsto
| l = a^2
| o = <
| r = a b
| c =
}}
{{eqn | ll= ... | Let $a, b \in \R$ be [[Definition:Real Number|real numbers]] such that $0 < a < b$.
Let $\map G {a, b}$ denote the [[Definition:Geometric Mean|geometric mean]] of $a$ and $b$.
Then:
:$a < \map G {a, b} < b$ | By definition of [[Definition:Geometric Mean|geometric mean]]:
:$\map G {a, b} := \sqrt {a b}$
where $\sqrt {a b}$ specifically denotes the [[Definition:Positive Square Root|positive square root]] of $a$ and $b$.
Thus:
{{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c =
}}
{{eqn | ll= \leadsto
... | Geometric Mean of two Positive Real Numbers is Between them | https://proofwiki.org/wiki/Geometric_Mean_of_two_Positive_Real_Numbers_is_Between_them | https://proofwiki.org/wiki/Geometric_Mean_of_two_Positive_Real_Numbers_is_Between_them | [
"Geometric Mean"
] | [
"Definition:Real Number",
"Definition:Geometric Mean"
] | [
"Definition:Geometric Mean",
"Definition:Square Root/Positive",
"Category:Geometric Mean"
] |
proofwiki-15979 | Geometric Mean of Reciprocals is Reciprocal of Geometric Mean | Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers.
Let $G_n$ denote the geometric mean of $x_1, x_2, \ldots, x_n$.
Let ${G_n}'$ denote the geometric mean of their reciprocals $\dfrac 1 {x_1}, \dfrac 1 {x_2}, \ldots, \dfrac 1 {x_n}$.
Then:
:${G_n}' = \dfrac 1 {G_n}$ | {{begin-eqn}}
{{eqn | l = {G_n}'
| r = \paren {\prod_{1 \mathop = k}^n \dfrac 1 {x_k} }^{1/n}
| c = {{Defof|Geometric Mean}}
}}
{{eqn | r = \paren {\dfrac 1 {\prod_{1 \mathop = k}^n x_k} }^{1/n}
| c =
}}
{{eqn | r = \dfrac 1 {\paren {\prod_{1 \mathop = k}^n x_k}^{1/n} }
| c =
}}
{{eqn | r = \d... | Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
Let $G_n$ denote the [[Definition:Geometric Mean|geometric mean]] of $x_1, x_2, \ldots, x_n$.
Let ${G_n}'$ denote the [[Definition:Geometric Mean|geometric mean]] of their [[Definition:Reciprocal|r... | {{begin-eqn}}
{{eqn | l = {G_n}'
| r = \paren {\prod_{1 \mathop = k}^n \dfrac 1 {x_k} }^{1/n}
| c = {{Defof|Geometric Mean}}
}}
{{eqn | r = \paren {\dfrac 1 {\prod_{1 \mathop = k}^n x_k} }^{1/n}
| c =
}}
{{eqn | r = \dfrac 1 {\paren {\prod_{1 \mathop = k}^n x_k}^{1/n} }
| c =
}}
{{eqn | r = \d... | Geometric Mean of Reciprocals is Reciprocal of Geometric Mean | https://proofwiki.org/wiki/Geometric_Mean_of_Reciprocals_is_Reciprocal_of_Geometric_Mean | https://proofwiki.org/wiki/Geometric_Mean_of_Reciprocals_is_Reciprocal_of_Geometric_Mean | [
"Geometric Mean",
"Reciprocals"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Geometric Mean",
"Definition:Geometric Mean",
"Definition:Reciprocal"
] | [
"Category:Geometric Mean",
"Category:Reciprocals"
] |
proofwiki-15980 | GM-HM Inequality | Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers.
Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$.
Let $H_n$ be the harmonic mean of $x_1, x_2, \ldots, x_n$.
Then:
:$G_n \ge H_n$ | Let ${G_n}'$ denotes the geometric mean of the reciprocals of $x_1, x_2, \ldots, x_n$.
By definition of harmonic mean, we have that:
:$\ds \dfrac 1 {H_n} := \frac 1 n \paren {\sum_{k \mathop = 1}^n \frac 1 {x_k} }$
That is, $\dfrac 1 {H_n}$ is the arithmetic mean of the reciprocals of $x_1, x_2, \ldots, x_n$.
Then:
{{b... | Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
Let $G_n$ be the [[Definition:Geometric Mean|geometric mean]] of $x_1, x_2, \ldots, x_n$.
Let $H_n$ be the [[Definition:Harmonic Mean|harmonic mean]] of $x_1, x_2, \ldots, x_n$.
Then:
:$G_n \ge H... | Let ${G_n}'$ denotes the [[Definition:Geometric Mean|geometric mean]] of the [[Definition:Reciprocal|reciprocals]] of $x_1, x_2, \ldots, x_n$.
By definition of [[Definition:Harmonic Mean|harmonic mean]], we have that:
:$\ds \dfrac 1 {H_n} := \frac 1 n \paren {\sum_{k \mathop = 1}^n \frac 1 {x_k} }$
That is, $\dfrac... | GM-HM Inequality/Proof 1 | https://proofwiki.org/wiki/GM-HM_Inequality | https://proofwiki.org/wiki/GM-HM_Inequality/Proof_1 | [
"GM-HM Inequality",
"Inequalities",
"Geometric Mean",
"Harmonic Mean"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Geometric Mean",
"Definition:Harmonic Mean"
] | [
"Definition:Geometric Mean",
"Definition:Reciprocal",
"Definition:Harmonic Mean",
"Definition:Arithmetic Mean",
"Definition:Reciprocal",
"Cauchy's Mean Theorem",
"Geometric Mean of Reciprocals is Reciprocal of Geometric Mean",
"Reciprocal Function is Strictly Decreasing"
] |
proofwiki-15981 | GM-HM Inequality | Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be strictly positive real numbers.
Let $G_n$ be the geometric mean of $x_1, x_2, \ldots, x_n$.
Let $H_n$ be the harmonic mean of $x_1, x_2, \ldots, x_n$.
Then:
:$G_n \ge H_n$ | For $p \in \R$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$'''.
By definition of Hölder Mean with $p = 0$:
:$\map {M_0} {x_1, x_2, \ldots, x_n} = \map G {x_1, x_2, \ldots, x_n}$
From Hölder Mean for Exponent -1 is Harmonic Mean:
:$\map {M_{-1} } {x_1, ... | Let $x_1, x_2, \ldots, x_n \in \R_{> 0}$ be [[Definition:Strictly Positive Real Number|strictly positive real numbers]].
Let $G_n$ be the [[Definition:Geometric Mean|geometric mean]] of $x_1, x_2, \ldots, x_n$.
Let $H_n$ be the [[Definition:Harmonic Mean|harmonic mean]] of $x_1, x_2, \ldots, x_n$.
Then:
:$G_n \ge H... | For $p \in \R$, let $\map {M_p} {x_1, x_2, \ldots, x_n}$ denote the '''[[Definition:Hölder Mean|Hölder mean with exponent $p$ of $x_1, x_2, \ldots, x_n$]]'''.
By definition of [[Definition:Hölder Mean with Zero Exponent|Hölder Mean with $p = 0$]]:
:$\map {M_0} {x_1, x_2, \ldots, x_n} = \map G {x_1, x_2, \ldots, x_n}$
... | GM-HM Inequality/Proof 2 | https://proofwiki.org/wiki/GM-HM_Inequality | https://proofwiki.org/wiki/GM-HM_Inequality/Proof_2 | [
"GM-HM Inequality",
"Inequalities",
"Geometric Mean",
"Harmonic Mean"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Geometric Mean",
"Definition:Harmonic Mean"
] | [
"Definition:Hölder Mean",
"Definition:Hölder Mean/Zero Exponent",
"Hölder Mean for Exponent -1 is Harmonic Mean",
"Inequality of Hölder Means"
] |
proofwiki-15982 | Harmonic Mean of two Positive Real Numbers is Between them | Let $a, b \in \R_{\gt 0}$ be (strictly) positive real numbers such that $a < b$.
Let $\map H {a, b}$ denote the harmonic mean of $a$ and $b$.
Then:
:$a < \map H {a, b} < b$ | By definition of harmonic mean:
:$\dfrac 1 {\map H {a, b} } := \dfrac 1 2 \paren {\dfrac 1 a + \dfrac 1 b}$
Thus:
{{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = \dfrac 1 b
| o = <
| r = \dfrac 1 a
| c = Reciprocal Function is Stri... | Let $a, b \in \R_{\gt 0}$ be [[Definition:Strictly Positive Real Number|(strictly) positive real numbers]] such that $a < b$.
Let $\map H {a, b}$ denote the [[Definition:Harmonic Mean|harmonic mean]] of $a$ and $b$.
Then:
:$a < \map H {a, b} < b$ | By definition of [[Definition:Harmonic Mean|harmonic mean]]:
:$\dfrac 1 {\map H {a, b} } := \dfrac 1 2 \paren {\dfrac 1 a + \dfrac 1 b}$
Thus:
{{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = \dfrac 1 b
| o = <
| r = \dfrac 1 a
... | Harmonic Mean of two Positive Real Numbers is Between them | https://proofwiki.org/wiki/Harmonic_Mean_of_two_Positive_Real_Numbers_is_Between_them | https://proofwiki.org/wiki/Harmonic_Mean_of_two_Positive_Real_Numbers_is_Between_them | [
"Harmonic Mean"
] | [
"Definition:Strictly Positive/Real Number",
"Definition:Harmonic Mean"
] | [
"Definition:Harmonic Mean",
"Reciprocal Function is Strictly Decreasing",
"Definition:Arithmetic Mean",
"Arithmetic Mean of two Real Numbers is Between them",
"Reciprocal Function is Strictly Decreasing",
"Category:Harmonic Mean"
] |
proofwiki-15983 | Arithmetic Mean of two Real Numbers is Between them | Let $a, b \in \R_{\ne 0}$ be real numbers such that $a < b$.
Let $\map A {a, b}$ denote the arithmetic mean of $a$ and $b$.
Then:
:$a < \map A {a, b} < b$ | By definition of arithmetic mean:
:$\map A {a, b} := \dfrac {a + b} 2$
Thus:
{{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = 2 a
| o = <
| r = a + b
| c = adding $a$ to both sides
}}
{{eqn | ll= \leadsto
| l = a
| o = <... | Let $a, b \in \R_{\ne 0}$ be [[Definition:Real Number|real numbers]] such that $a < b$.
Let $\map A {a, b}$ denote the [[Definition:Arithmetic Mean|arithmetic mean]] of $a$ and $b$.
Then:
:$a < \map A {a, b} < b$ | By definition of [[Definition:Arithmetic Mean|arithmetic mean]]:
:$\map A {a, b} := \dfrac {a + b} 2$
Thus:
{{begin-eqn}}
{{eqn | l = a
| o = <
| r = b
| c = {{hypothesis}}
}}
{{eqn | ll= \leadsto
| l = 2 a
| o = <
| r = a + b
| c = [[Definition:Real Addition|adding]] $a$ t... | Arithmetic Mean of two Real Numbers is Between them | https://proofwiki.org/wiki/Arithmetic_Mean_of_two_Real_Numbers_is_Between_them | https://proofwiki.org/wiki/Arithmetic_Mean_of_two_Real_Numbers_is_Between_them | [
"Arithmetic Mean"
] | [
"Definition:Real Number",
"Definition:Arithmetic Mean"
] | [
"Definition:Arithmetic Mean",
"Definition:Addition/Real Numbers",
"Definition:Division/Field/Real Numbers",
"Definition:Addition/Real Numbers",
"Definition:Division/Field/Real Numbers",
"Category:Arithmetic Mean"
] |
proofwiki-15984 | Kurtosis of Normal Distribution | Let $X$ be a continuous random variable with a normal distribution with parameters $\mu$ and $\sigma^2$ for some $\mu \in \R$ and $\sigma \in \R_{> 0}$:
:$X \sim \Gaussian \mu {\sigma^2}$
Then the kurtosis $\alpha_4$ of $X$ is equal to $3$.
That is, $\Gaussian \mu {\sigma^2}$ is mesokurtic. | From the definition of kurtosis, we have:
:$\alpha_4 = \expect {\paren {\dfrac {X - \mu} \sigma}^4}$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Normal Distribution, we have:
:$\mu = \mu$
By Variance of Normal Distribution, we have:
:$\sigma = \sigma$
So:... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] with a [[Definition:Normal Distribution|normal distribution with parameters $\mu$ and $\sigma^2$]] for some $\mu \in \R$ and $\sigma \in \R_{> 0}$:
:$X \sim \Gaussian \mu {\sigma^2}$
Then the [[Definition:Kurtosis|kurtosis]] $\alpha_4$ ... | From the definition of [[Definition:Kurtosis|kurtosis]], we have:
:$\alpha_4 = \expect {\paren {\dfrac {X - \mu} \sigma}^4}$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Normal Distribution]], ... | Kurtosis of Normal Distribution | https://proofwiki.org/wiki/Kurtosis_of_Normal_Distribution | https://proofwiki.org/wiki/Kurtosis_of_Normal_Distribution | [
"Kurtosis",
"Normal Distribution"
] | [
"Definition:Random Variable/Continuous",
"Definition:Normal Distribution",
"Definition:Kurtosis",
"Definition:Mesokurtic"
] | [
"Definition:Kurtosis",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Normal Distribution",
"Variance of Normal Distribution",
"Kurtosis in terms of Non-Central Moments",
"Skewness of Normal Distribution/Proof 2",
"Variance of Normal Distribution/Proof 2",
"Definition:Mom... |
proofwiki-15985 | Raw Moment of Bernoulli Distribution | Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$.
Let $n$ be a strictly positive integer.
Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:
:$\expect {X^n} = p$ | From the definition of expectation:
:$\ds \expect {X^n} = \sum_{x \mathop \in \Img X} x^n \map \Pr {X = x}$
From the definition of the Bernoulli distribution:
:$\ds \expect {X^n} = 1^n \times p + 0^n \times \paren {1 - p} = p$
{{qed}} | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with a [[Definition:Bernoulli Distribution|Bernoulli distribution with parameter $p$]].
Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Then the $n$th [[Definition:Raw Moment|raw moment]] $\expect {X^n}$ o... | From the definition of [[Definition:Expectation|expectation]]:
:$\ds \expect {X^n} = \sum_{x \mathop \in \Img X} x^n \map \Pr {X = x}$
From the definition of the [[Definition:Bernoulli Distribution|Bernoulli distribution]]:
:$\ds \expect {X^n} = 1^n \times p + 0^n \times \paren {1 - p} = p$
{{qed}} | Raw Moment of Bernoulli Distribution/Proof 1 | https://proofwiki.org/wiki/Raw_Moment_of_Bernoulli_Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Bernoulli_Distribution/Proof_1 | [
"Bernoulli Distribution",
"Raw Moments",
"Raw Moment of Bernoulli Distribution"
] | [
"Definition:Random Variable/Discrete",
"Definition:Bernoulli Distribution",
"Definition:Strictly Positive/Integer",
"Definition:Raw Moment"
] | [
"Definition:Expectation",
"Definition:Bernoulli Distribution"
] |
proofwiki-15986 | Equivalence of Definitions of Local Basis | Let $T = \struct {S, \tau}$ be a topological space.
Let $x$ be an element of $S$.
{{TFAE|def = Local Basis}} | === Local Basis for Open Sets Implies Neighborhood Basis of Open Sets ===
{{:Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets}}{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x$ be an [[Definition:Element|element]] of $S$.
{{TFAE|def = Local Basis}} | === [[Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets|Local Basis for Open Sets Implies Neighborhood Basis of Open Sets]] ===
{{:Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets}}{{qed|lemma}} | Equivalence of Definitions of Local Basis | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Local_Basis | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Local_Basis | [
"Local Bases",
"Equivalence of Definitions of Local Basis"
] | [
"Definition:Topological Space",
"Definition:Element"
] | [
"Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets"
] |
proofwiki-15987 | Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets | Let $\BB$ be a set of open neighborhoods of $x$ such that:
:$\forall U \in \tau: x \in U \implies \exists H \in \BB: H \subseteq U$ | Let $N$ be a neighborhood of $x$.
Then there exists $U \in \tau$ such that $x \in U$ and $U \subseteq N$ by definition.
By assumption, there exists $H \in \BB$ such that $H \subseteq U$.
From Subset Relation is Transitive, $H \subseteq N$.
The result follows. | Let $\BB$ be a [[Definition:Set|set]] of [[Definition:Open Neighborhood of Point|open neighborhoods]] of $x$ such that:
:$\forall U \in \tau: x \in U \implies \exists H \in \BB: H \subseteq U$ | Let $N$ be a [[Definition:Neighborhood of Point|neighborhood]] of $x$.
Then there exists $U \in \tau$ such that $x \in U$ and $U \subseteq N$ by definition.
By assumption, there exists $H \in \BB$ such that $H \subseteq U$.
From [[Subset Relation is Transitive]], $H \subseteq N$.
The result follows. | Equivalence of Definitions of Local Basis/Local Basis for Open Sets Implies Neighborhood Basis of Open Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Local_Basis/Local_Basis_for_Open_Sets_Implies_Neighborhood_Basis_of_Open_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Local_Basis/Local_Basis_for_Open_Sets_Implies_Neighborhood_Basis_of_Open_Sets | [
"Equivalence of Definitions of Local Basis"
] | [
"Definition:Set",
"Definition:Open Neighborhood/Point"
] | [
"Definition:Neighborhood (Topology)/Point",
"Subset Relation is Transitive"
] |
proofwiki-15988 | Equivalence of Definitions of Local Basis/Neighborhood Basis of Open Sets Implies Local Basis for Open Sets | Let $\BB$ be a set of open neighborhoods of $x$ such that:
:every neighborhood of $x$ contains a set in $\BB$. | Let $U \in \tau$ such that $x \in U$.
From Set is Open iff Neighborhood of all its Points then $U$ is a neighborhood of $x$.
By assumption, there exists $H \in \BB$ such that $H \subseteq U$.
The result follows. | Let $\BB$ be a [[Definition:Set|set]] of [[Definition:Open Neighborhood of Point|open neighborhoods]] of $x$ such that:
:every [[Definition:Neighborhood of Point|neighborhood]] of $x$ contains a set in $\BB$. | Let $U \in \tau$ such that $x \in U$.
From [[Set is Open iff Neighborhood of all its Points]] then $U$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$.
By assumption, there exists $H \in \BB$ such that $H \subseteq U$.
The result follows. | Equivalence of Definitions of Local Basis/Neighborhood Basis of Open Sets Implies Local Basis for Open Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Local_Basis/Neighborhood_Basis_of_Open_Sets_Implies_Local_Basis_for_Open_Sets | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Local_Basis/Neighborhood_Basis_of_Open_Sets_Implies_Local_Basis_for_Open_Sets | [
"Equivalence of Definitions of Local Basis"
] | [
"Definition:Set",
"Definition:Open Neighborhood/Point",
"Definition:Neighborhood (Topology)/Point"
] | [
"Set is Open iff Neighborhood of all its Points",
"Definition:Neighborhood (Topology)/Point"
] |
proofwiki-15989 | Local Basis Generated from Neighborhood Basis | Let $T = \struct {S, \tau}$ be a topological space.
Let $x$ be an element of $S$.
Let $\BB$ be a neighborhood basis of $x$.
For any subset $A \subseteq S$, let $A^\circ$ denote the interior of $A$.
Then the set:
:$\BB' = \set {H^\circ: H \in \BB}$
is a local basis of $x$. | First it must be shown that $\BB'$ is a set of open neighborhoods of $x$.
From the definition of the interior of a subset, $\BB'$ is a set of open sets.
Let $H \in \BB$.
By assumption, $H$ is a neighborhood of $x$.
From the definition of a neighborhood:
:$\exists U \in \tau : x \in U \subseteq H$
From the definition o... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x$ be an [[Definition:Element|element]] of $S$.
Let $\BB$ be a [[Definition:Neighborhood Basis|neighborhood basis]] of $x$.
For any [[Definition:Subset|subset]] $A \subseteq S$, let $A^\circ$ denote the [[Definition:Interior (T... | First it must be shown that $\BB'$ is a [[Definition:Set|set]] of [[Definition:Open Neighborhood of Point|open neighborhoods]] of $x$.
From the definition of the [[Definition:Interior (Topology)|interior]] of a [[Definition:Subset|subset]], $\BB'$ is a [[Definition:Set|set]] of [[Definition:Open Set|open sets]].
Let ... | Local Basis Generated from Neighborhood Basis | https://proofwiki.org/wiki/Local_Basis_Generated_from_Neighborhood_Basis | https://proofwiki.org/wiki/Local_Basis_Generated_from_Neighborhood_Basis | [
"Local Bases",
"Neighborhood Bases"
] | [
"Definition:Topological Space",
"Definition:Element",
"Definition:Neighborhood Basis",
"Definition:Subset",
"Definition:Interior (Topology)",
"Definition:Local Basis"
] | [
"Definition:Set",
"Definition:Open Neighborhood/Point",
"Definition:Interior (Topology)",
"Definition:Subset",
"Definition:Set",
"Definition:Open Set",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Interior (Topology)",
"Definition:Subset",
"... |
proofwiki-15990 | Arcsine of Zero is Zero | :$\arcsin 0 = 0$
where $\arcsin$ is the arcsine function. | By definition, $\arcsin$ is the inverse of the restriction of the sine function to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Therefore, if:
:$\sin x = 0$
and $-\dfrac \pi 2 \le x \le \dfrac \pi 2$, then $\arcsin 0 = x$.
From Sine of Zero is Zero, we have that:
:$\sin 0 = 0$
We have $-\dfrac \pi 2 < 0 < \dfrac \pi... | :$\arcsin 0 = 0$
where $\arcsin$ is the [[Definition:Real Arcsine|arcsine function]]. | By definition, $\arcsin$ is the [[Definition:Inverse Mapping|inverse]] of the [[Definition:Restriction of Mapping|restriction]] of the [[Definition:Sine Function|sine function]] to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Therefore, if:
:$\sin x = 0$
and $-\dfrac \pi 2 \le x \le \dfrac \pi 2$, then $\arcsin 0 =... | Arcsine of Zero is Zero | https://proofwiki.org/wiki/Arcsine_of_Zero_is_Zero | https://proofwiki.org/wiki/Arcsine_of_Zero_is_Zero | [
"Arcsine Function"
] | [
"Definition:Inverse Sine/Real/Arcsine"
] | [
"Definition:Inverse Mapping",
"Definition:Restriction/Mapping",
"Definition:Sine",
"Sine of Zero is Zero",
"Category:Arcsine Function"
] |
proofwiki-15991 | Arcsine of One is Half Pi | :$\arcsin 1 = \dfrac \pi 2$
where $\arcsin$ is the arcsine function. | By definition, $\arcsin$ is the inverse of the restriction of the sine function to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Therefore, if:
:$\sin x = 1$
and $-\dfrac \pi 2 \le x \le \dfrac \pi 2$, then $\arcsin 1 = x$.
From Sine of Right Angle, we have that:
:$\sin \dfrac \pi 2 = 1$
We therefore have:
:$\arcsin ... | :$\arcsin 1 = \dfrac \pi 2$
where $\arcsin$ is the [[Definition:Real Arcsine|arcsine function]]. | By definition, $\arcsin$ is the [[Definition:Inverse Mapping|inverse]] of the [[Definition:Restriction of Mapping|restriction]] of the [[Definition:Sine Function|sine function]] to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Therefore, if:
:$\sin x = 1$
and $-\dfrac \pi 2 \le x \le \dfrac \pi 2$, then $\arcsin 1 =... | Arcsine of One is Half Pi | https://proofwiki.org/wiki/Arcsine_of_One_is_Half_Pi | https://proofwiki.org/wiki/Arcsine_of_One_is_Half_Pi | [
"Arcsine Function"
] | [
"Definition:Inverse Sine/Real/Arcsine"
] | [
"Definition:Inverse Mapping",
"Definition:Restriction/Mapping",
"Definition:Sine",
"Sine of Right Angle",
"Category:Arcsine Function"
] |
proofwiki-15992 | Real Sequence (1 + x over n)^n is Convergent | The sequence $\sequence {s_n}$ defined as:
:$s_n = \paren {1 + \dfrac x n}^n$
is convergent. | From Cauchy's Mean Theorem:
:$(1): \quad \ds \paren {\prod_{k \mathop = 1}^n a_k}^{1/n} \le \frac 1 n \paren {\sum_{k \mathop = 1}^n a_k}$
for $r_1, r_2, \ldots, r_n$.
Setting:
:$a_1 = a_2 = \ldots = a_{n - 1} := 1 + \dfrac x {n - 1}$
and:
:$a_n = 1$
Substituting for $a_1, a_2, \ldots, a_n$ into $(1)$ gives:
{{begin-eq... | The [[Definition:Real Sequence|sequence]] $\sequence {s_n}$ defined as:
:$s_n = \paren {1 + \dfrac x n}^n$
is [[Definition:Convergent Real Sequence|convergent]]. | From [[Cauchy's Mean Theorem]]:
:$(1): \quad \ds \paren {\prod_{k \mathop = 1}^n a_k}^{1/n} \le \frac 1 n \paren {\sum_{k \mathop = 1}^n a_k}$
for $r_1, r_2, \ldots, r_n$.
Setting:
:$a_1 = a_2 = \ldots = a_{n - 1} := 1 + \dfrac x {n - 1}$
and:
:$a_n = 1$
Substituting for $a_1, a_2, \ldots, a_n$ into $(1)$ gives:
{... | Real Sequence (1 + x over n)^n is Convergent | https://proofwiki.org/wiki/Real_Sequence_(1_+_x_over_n)^n_is_Convergent | https://proofwiki.org/wiki/Real_Sequence_(1_+_x_over_n)^n_is_Convergent | [
"Exponential Function"
] | [
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Cauchy's Mean Theorem",
"Definition:Positive/Real Number",
"Definition:Increasing/Sequence/Real Sequence",
"Equivalence of Definitions of Real Exponential Function/Limit of Sequence implies Power Series Expansion",
"Definition:Strictly Increasing/Sequence/Real Sequence",
"Definition:Bounded Above Sequenc... |
proofwiki-15993 | Raw Moment of Poisson Distribution | Let $X$ be a discrete random variable with the Poisson distribution with parameter $\lambda$.
Let $n$ be a strictly positive integer.
Then the $n$th raw moment $\expect {X^n}$ of $X$ is given by:
:$\ds \expect {X^n} = \sum_{k \mathop = 0}^n \lambda^k {n \brace k}$
where $\ds {n \brace k}$ is a Stirling number of the ... | {{ProofWanted}}
Category:Poisson Distribution
Category:Raw Moments
tmu1rjvgmzdbq01np4exsclvco4henz | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with the [[Definition:Poisson Distribution|Poisson distribution with parameter $\lambda$]].
Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Then the $n$th [[Definition:Raw Moment|raw moment]] $\expect {X^n... | {{ProofWanted}}
[[Category:Poisson Distribution]]
[[Category:Raw Moments]]
tmu1rjvgmzdbq01np4exsclvco4henz | Raw Moment of Poisson Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Poisson_Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Poisson_Distribution | [
"Poisson Distribution",
"Raw Moments"
] | [
"Definition:Random Variable/Discrete",
"Definition:Poisson Distribution",
"Definition:Strictly Positive/Integer",
"Definition:Raw Moment",
"Definition:Stirling Numbers of the Second Kind"
] | [
"Category:Poisson Distribution",
"Category:Raw Moments"
] |
proofwiki-15994 | Raw Moment of Exponential Distribution | Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{>0}$.
Let $n$ be a strictly positive integer.
Then the $n$th raw moment of $X$ is given by:
:$\expect {X^n} = n! \beta^n$ | From Moment Generating Function of Exponential Distribution, the moment generating function of $X$ is given by:
:$\map {M_X} t = \dfrac 1 {1 - \beta t}$
By Moment in terms of Moment Generating Function:
:$\expect {X^n} = \map {M^{\paren n}_X} 0$
We have:
{{begin-eqn}}
{{eqn | l = \map {M^{\paren n}_X} t
| r = \frac {... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] of the [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$ for some $\beta \in \R_{>0}$.
Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Then the $n$th [[Definition:... | From [[Moment Generating Function of Exponential Distribution]], the [[Definition:Moment Generating Function|moment generating function]] of $X$ is given by:
:$\map {M_X} t = \dfrac 1 {1 - \beta t}$
By [[Moment in terms of Moment Generating Function]]:
:$\expect {X^n} = \map {M^{\paren n}_X} 0$
We have:
{{begin-e... | Raw Moment of Exponential Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Exponential_Distribution | https://proofwiki.org/wiki/Raw_Moment_of_Exponential_Distribution | [
"Exponential Distribution",
"Raw Moments"
] | [
"Definition:Random Variable/Continuous",
"Definition:Exponential Distribution",
"Definition:Strictly Positive/Integer",
"Definition:Raw Moment"
] | [
"Moment Generating Function of Exponential Distribution",
"Definition:Moment Generating Function",
"Moment in terms of Moment Generating Function",
"Nth Derivative of Reciprocal of Mth Power/Corollary",
"Derivative of Composite Function",
"Category:Exponential Distribution",
"Category:Raw Moments"
] |
proofwiki-15995 | Central Moment of Exponential Distribution | Let $X$ be a continuous random variable of the exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$
Let $n$ be a strictly positive integer.
Then the $n$th central moment $\mu_n$ of $X$ is given by:
:$\ds \mu_n = n! \beta^n \sum_{k \mathop = 0}^n \frac {\paren {-1}^k} {k!}$ | From definition of central moment we have:
:$\mu_n = \expect {\paren {x - \mu}^n}$
By Expectation of Exponential Distribution we have:
:$\mu = \beta$
So:
{{begin-eqn}}
{{eqn | l = \mu_2
| r = \expect {\sum_{k \mathop = 0}^n \binom n k X^{n - k} \paren {-\beta}^k}
| c = Binomial Theorem
}}
{{eqn | r = \sum_... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]] of the [[Definition:Exponential Distribution|exponential distribution]] with parameter $\beta$ for some $\beta \in \R_{> 0}$
Let $n$ be a [[Definition:Strictly Positive Integer|strictly positive integer]].
Then the $n$th [[Definition:... | From definition of [[Definition:Central Moment|central moment]] we have:
:$\mu_n = \expect {\paren {x - \mu}^n}$
By [[Expectation of Exponential Distribution]] we have:
:$\mu = \beta$
So:
{{begin-eqn}}
{{eqn | l = \mu_2
| r = \expect {\sum_{k \mathop = 0}^n \binom n k X^{n - k} \paren {-\beta}^k}
| c = ... | Central Moment of Exponential Distribution | https://proofwiki.org/wiki/Central_Moment_of_Exponential_Distribution | https://proofwiki.org/wiki/Central_Moment_of_Exponential_Distribution | [
"Exponential Distribution",
"Central Moments"
] | [
"Definition:Random Variable/Continuous",
"Definition:Exponential Distribution",
"Definition:Strictly Positive/Integer",
"Definition:Central Moment"
] | [
"Definition:Central Moment",
"Expectation of Exponential Distribution",
"Binomial Theorem",
"Expectation is Linear",
"Raw Moment of Exponential Distribution",
"Category:Exponential Distribution",
"Category:Central Moments"
] |
proofwiki-15996 | Excess Kurtosis of Bernoulli Distribution | Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$.
Then the excess kurtosis $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac {1 - 6 p q} {p q}$
where $q = 1 - p$. | From the definition of excess kurtosis, we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Bernoulli Distribution, we have:
:$\mu = p$
By Variance of Bernoulli Distribution, we have:
:$\sigma... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]] with a [[Definition:Bernoulli Distribution|Bernoulli distribution with parameter $p$]].
Then the [[Definition:Excess Kurtosis|excess kurtosis]] $\gamma_2$ of $X$ is given by:
:$\gamma_2 = \dfrac {1 - 6 p q} {p q}$
where $q = 1 - p$. | From the definition of [[Definition:Excess Kurtosis|excess kurtosis]], we have:
:$\gamma_2 = \expect {\paren {\dfrac {X - \mu} \sigma}^4} - 3$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Berno... | Excess Kurtosis of Bernoulli Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Bernoulli_Distribution | https://proofwiki.org/wiki/Excess_Kurtosis_of_Bernoulli_Distribution | [
"Kurtosis",
"Bernoulli Distribution"
] | [
"Definition:Random Variable/Discrete",
"Definition:Bernoulli Distribution",
"Definition:Excess Kurtosis"
] | [
"Definition:Excess Kurtosis",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Bernoulli Distribution",
"Variance of Bernoulli Distribution",
"Kurtosis in terms of Non-Central Moments",
"Raw Moment of Bernoulli Distribution",
"Category:Kurtosis",
"Category:Bernoulli Distrib... |
proofwiki-15997 | Product of Limits of Real Sequences (1 + x over n)^n and (1 - x over n)^n equals 1 | Let $\sequence {a_n}$ be the sequence defined as:
:$a_n = \paren {1 + \dfrac x n}^n$
Let $\sequence {b_n}$ be the sequence defined as:
:$b_n = \paren {1 - \dfrac x n}^n$
Then the product of the limits of $\sequence {a_n}$ and $\sequence {b_n}$ equals $1$ | From Real Sequence $\paren {1 + \dfrac x n}^n$ is Convergent, $\sequence {a_n}$ is convergent.
Setting $x \to -x$, it follows that $\sequence {\paren {1 + \dfrac {\paren {-x} } n}^n} = \sequence {b_n}$ is also convergent.
Then:
{{begin-eqn}}
{{eqn | l = \paren {1 - \dfrac x n}^n \paren {1 - \dfrac x n}^n
| r = \p... | Let $\sequence {a_n}$ be the [[Definition:Real Sequence|sequence]] defined as:
:$a_n = \paren {1 + \dfrac x n}^n$
Let $\sequence {b_n}$ be the [[Definition:Real Sequence|sequence]] defined as:
:$b_n = \paren {1 - \dfrac x n}^n$
Then the [[Definition:Real Multiplication|product]] of the [[Definition:Limit of Real S... | From [[Real Sequence (1 + x over n)^n is Convergent|Real Sequence $\paren {1 + \dfrac x n}^n$ is Convergent]], $\sequence {a_n}$ is [[Definition:Convergent Real Sequence|convergent]].
Setting $x \to -x$, it follows that $\sequence {\paren {1 + \dfrac {\paren {-x} } n}^n} = \sequence {b_n}$ is also [[Definition:Converg... | Product of Limits of Real Sequences (1 + x over n)^n and (1 - x over n)^n equals 1 | https://proofwiki.org/wiki/Product_of_Limits_of_Real_Sequences_(1_+_x_over_n)^n_and_(1_-_x_over_n)^n_equals_1 | https://proofwiki.org/wiki/Product_of_Limits_of_Real_Sequences_(1_+_x_over_n)^n_and_(1_-_x_over_n)^n_equals_1 | [
"Examples of Convergent Real Sequences"
] | [
"Definition:Real Sequence",
"Definition:Real Sequence",
"Definition:Multiplication/Real Numbers",
"Definition:Limit of Sequence/Real Numbers"
] | [
"Real Sequence (1 + x over n)^n is Convergent",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers",
"Difference of Two Squares",
"Real Sequence (1 + x over n)^n is Convergent",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Positive/Real Number",
"... |
proofwiki-15998 | Farey Sequence is not Convergent | Consider the Farey sequence:
:$F = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$
$F$ is not convergent. | We have the following subsequences of $F$ which are all convergent to a different limit:
:$\dfrac 1 2, \dfrac 2 4, \dfrac 3 6, \dfrac 4 8 \to \dfrac 1 2$ as $n \to \infty$
:$\dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \dfrac 1 5 \to 0$ as $n \to \infty$
:$\dfrac 1 2, \dfrac 2 3, \dfrac 3 4, \dfrac 4 5 \to 1$ as $n \to \infty$
... | Consider the [[Definition:Farey Sequence|Farey sequence]]:
:$F = \dfrac 1 2, \dfrac 1 3, \dfrac 2 3, \dfrac 1 4, \dfrac 2 4, \dfrac 3 4, \dfrac 1 5, \dfrac 2 5, \dfrac 3 5, \dfrac 4 5, \dfrac 1 6, \ldots$
$F$ is not [[Definition:Convergent Real Sequence|convergent]]. | We have the following [[Definition:Subsequence|subsequences]] of $F$ which are all [[Definition:Convergent Real Sequence|convergent]] to a different [[Definition:Limit of Real Sequence|limit]]:
:$\dfrac 1 2, \dfrac 2 4, \dfrac 3 6, \dfrac 4 8 \to \dfrac 1 2$ as $n \to \infty$
:$\dfrac 1 2, \dfrac 1 3, \dfrac 1 4, \df... | Farey Sequence is not Convergent | https://proofwiki.org/wiki/Farey_Sequence_is_not_Convergent | https://proofwiki.org/wiki/Farey_Sequence_is_not_Convergent | [
"Farey Sequences",
"Divergent Sequences"
] | [
"Definition:Farey Sequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Subsequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Limit of Sequence/Real Numbers",
"Limit of Subsequence equals Limit of Sequence/Real Numbers",
"Definition:Convergent Sequence/Real Numbers"
] |
proofwiki-15999 | Skewness of Gamma Distribution | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
Then the skewness $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac 2 {\sqrt \alpha}$ | From Skewness in terms of Non-Central Moments, we have:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the expectation of $X$.
:$\sigma$ is the standard deviation of $X$.
By Expectation of Gamma Distribution, we have:
:$\mu = \dfrac \alpha \beta$
By Variance of Gamma Distr... | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the [[Definition:Gamma Distribution|Gamma distribution]].
Then the [[Definition:Skewness|skewness]] $\gamma_1$ of $X$ is given by:
:$\gamma_1 = \dfrac 2 {\sqrt \alpha}$ | From [[Skewness in terms of Non-Central Moments]], we have:
:$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
where:
:$\mu$ is the [[Definition:Expectation|expectation]] of $X$.
:$\sigma$ is the [[Definition:Standard Deviation|standard deviation]] of $X$.
By [[Expectation of Gamma Distribut... | Skewness of Gamma Distribution/Proof 1 | https://proofwiki.org/wiki/Skewness_of_Gamma_Distribution | https://proofwiki.org/wiki/Skewness_of_Gamma_Distribution/Proof_1 | [
"Gamma Distribution",
"Skewness",
"Skewness of Gamma Distribution"
] | [
"Definition:Gamma Distribution",
"Definition:Skewness"
] | [
"Skewness in terms of Non-Central Moments",
"Definition:Expectation",
"Definition:Standard Deviation",
"Expectation of Gamma Distribution",
"Variance of Gamma Distribution",
"Moment in terms of Moment Generating Function",
"Definition:Moment Generating Function",
"Moment Generating Function of Gamma D... |
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