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-14200 | Definite Integral from 0 to a of Reciprocal of Root of a Squared minus x Squared | :$\ds \int_0^a \dfrac {\d x} {\sqrt {a^2 - x^2} } = \frac \pi 2$
for $a > 0$. | {{begin-eqn}}
{{eqn | l = \int_0^a \dfrac {\d x} {\sqrt {a^2 - x^2} }
| r = \int_0^{\mathop \to a} \dfrac {\d x} {\sqrt {a^2 - x^2} }
| c = as $\dfrac 1 {\sqrt {a^2 - x^2} }$ does not exist for $x = a$
}}
{{eqn | r = \lim_{\gamma \mathop \to a} \int_0^\gamma \dfrac {\d x} {\sqrt {a^2 - x^2} }
| c = {{... | :$\ds \int_0^a \dfrac {\d x} {\sqrt {a^2 - x^2} } = \frac \pi 2$
for $a > 0$. | {{begin-eqn}}
{{eqn | l = \int_0^a \dfrac {\d x} {\sqrt {a^2 - x^2} }
| r = \int_0^{\mathop \to a} \dfrac {\d x} {\sqrt {a^2 - x^2} }
| c = as $\dfrac 1 {\sqrt {a^2 - x^2} }$ does not exist for $x = a$
}}
{{eqn | r = \lim_{\gamma \mathop \to a} \int_0^\gamma \dfrac {\d x} {\sqrt {a^2 - x^2} }
| c = {{... | Definite Integral from 0 to a of Reciprocal of Root of a Squared minus x Squared/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_a_of_Reciprocal_of_Root_of_a_Squared_minus_x_Squared | https://proofwiki.org/wiki/Definite_Integral_from_0_to_a_of_Reciprocal_of_Root_of_a_Squared_minus_x_Squared/Proof_1 | [
"Definite Integral from 0 to a of Reciprocal of Root of a Squared minus x Squared",
"Examples of Definite Integrals"
] | [] | [
"Primitive of Reciprocal of Root of a squared minus x squared/Arcsine Form",
"Arcsine of Zero is Zero",
"Arcsine of One is Half Pi"
] |
proofwiki-14201 | Universal Property of Free Abelian Group on Set | Let $S$ be a set.
Let $\struct {\Z^{\paren S}, \iota}$ be the free abelian group on $S$.
Let $\struct {G, +}$ be an abelian group.
Let $f: S \to G$ be a mapping.
Then there exists a unique group homomorphism $g : \Z^{\paren S} \to G$ with $g \circ \iota = f$:
:$\xymatrix{S \ar[d]_\iota \ar[r]^{\forall f} & G\\ \Z^{\par... | For $x \in S$, the characteristic map $\phi_x$ of $x$ is defined as:
:$\map {\phi_x} s = \begin{cases} 1 & \quad \text{ if } s = x \\ 0 & \quad \text{ if } s \ne x \end{cases}$
The maps $\set {\phi_x \ : \ x \in S}$ form a basis for $\Z^{\paren S}$.
If $\phi \in \Z^{\paren S}$, then it can be expressed uniquely as a fi... | Let $S$ be a [[Definition:Set|set]].
Let $\struct {\Z^{\paren S}, \iota}$ be the [[Definition:Free Abelian Group on Set|free abelian group]] on $S$.
Let $\struct {G, +}$ be an [[Definition:Abelian Group|abelian group]].
Let $f: S \to G$ be a [[Definition:Mapping|mapping]].
Then there exists a [[Definition:Unique|u... | For $x \in S$, the [[Definition:Characteristic Mapping|characteristic map]] $\phi_x$ of $x$ is defined as:
:$\map {\phi_x} s = \begin{cases} 1 & \quad \text{ if } s = x \\ 0 & \quad \text{ if } s \ne x \end{cases}$
The maps $\set {\phi_x \ : \ x \in S}$ form a basis for $\Z^{\paren S}$.
If $\phi \in \Z^{\paren S}$, th... | Universal Property of Free Abelian Group on Set | https://proofwiki.org/wiki/Universal_Property_of_Free_Abelian_Group_on_Set | https://proofwiki.org/wiki/Universal_Property_of_Free_Abelian_Group_on_Set | [
"Universal Properties",
"Abelian Groups"
] | [
"Definition:Set",
"Definition:Free Abelian Group on Set",
"Definition:Abelian Group",
"Definition:Mapping",
"Definition:Unique",
"Definition:Group Homomorphism"
] | [
"Definition:Characteristic Function (Set Theory)",
"Definition:Group Homomorphism",
"Definition:Group Homomorphism"
] |
proofwiki-14202 | Definite Integral from 0 to a of Root of a Squared minus x Squared | :$\ds \int_0^a \sqrt {a^2 - x^2} \rd x = \frac {\pi a^2} 4$
for $a > 0$. | {{begin-eqn}}
{{eqn | l = \int_0^a \sqrt {a^2 - x^2} \rd x
| r = \intlimits {\frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a} 0 a
| c = Primitive of $\sqrt {a^2 - x^2}$
}}
{{eqn | r = \paren {\frac {a \sqrt {a^2 - a^2} } 2 + \frac {a^2} 2 \arcsin \frac a a} - \paren {\frac {0 \sqrt {a^2 - x... | :$\ds \int_0^a \sqrt {a^2 - x^2} \rd x = \frac {\pi a^2} 4$
for $a > 0$. | {{begin-eqn}}
{{eqn | l = \int_0^a \sqrt {a^2 - x^2} \rd x
| r = \intlimits {\frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a} 0 a
| c = [[Primitive of Root of a squared minus x squared|Primitive of $\sqrt {a^2 - x^2}$]]
}}
{{eqn | r = \paren {\frac {a \sqrt {a^2 - a^2} } 2 + \frac {a^2} 2 \... | Definite Integral from 0 to a of Root of a Squared minus x Squared/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_a_of_Root_of_a_Squared_minus_x_Squared | https://proofwiki.org/wiki/Definite_Integral_from_0_to_a_of_Root_of_a_Squared_minus_x_Squared/Proof_1 | [
"Definite Integral from 0 to a of Root of a Squared minus x Squared",
"Examples of Definite Integrals"
] | [] | [
"Primitive of Root of a squared minus x squared",
"Sine of Zero is Zero",
"Sine of Right Angle"
] |
proofwiki-14203 | Definite Integral from 0 to a of Root of a Squared minus x Squared | :$\ds \int_0^a \sqrt {a^2 - x^2} \rd x = \frac {\pi a^2} 4$
for $a > 0$. | {{begin-eqn}}
{{eqn | l = \int_0^a \sqrt {a^2 - x^2} \rd x
| r = \frac {a^{1 + \frac 2 2} } 2 \frac {\map \Gamma {\frac 1 2} \map \Gamma {1 + \frac 1 2} } {\map \Gamma {\frac 1 2 + \frac 1 2 + 1} }
| c = Definite Integral from 0 to a of $x^m \paren {a^n - x^n}^p$
}}
{{eqn | r = \frac {a^2} 2 \frac {\map \Gamma {\fra... | :$\ds \int_0^a \sqrt {a^2 - x^2} \rd x = \frac {\pi a^2} 4$
for $a > 0$. | {{begin-eqn}}
{{eqn | l = \int_0^a \sqrt {a^2 - x^2} \rd x
| r = \frac {a^{1 + \frac 2 2} } 2 \frac {\map \Gamma {\frac 1 2} \map \Gamma {1 + \frac 1 2} } {\map \Gamma {\frac 1 2 + \frac 1 2 + 1} }
| c = [[Definite Integral from 0 to a of x^m by (a^n - x^n)^p|Definite Integral from 0 to a of $x^m \paren {a^n - x^n}^... | Definite Integral from 0 to a of Root of a Squared minus x Squared/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_a_of_Root_of_a_Squared_minus_x_Squared | https://proofwiki.org/wiki/Definite_Integral_from_0_to_a_of_Root_of_a_Squared_minus_x_Squared/Proof_2 | [
"Definite Integral from 0 to a of Root of a Squared minus x Squared",
"Examples of Definite Integrals"
] | [] | [
"Definite Integral from 0 to a of x^m by (a^n - x^n)^p",
"Gamma Difference Equation",
"Gamma Function Extends Factorial",
"Gamma Function of One Half"
] |
proofwiki-14204 | Definite Integral from 0 to a of x^m by (a^n - x^n)^p | :$\ds \int_0^a x^m \paren {a^n - x^n}^p \rd x = \frac {a^{m + 1 + n p} \, \map \Gamma {\frac {m + 1} n} \map \Gamma {p + 1} } {n \map \Gamma {\frac {m + 1} n + p + 1} }$ | {{begin-eqn}}
{{eqn | l = \int_0^a x^m \paren {a^n - x^n}^p \rd x
| r = a \int_0^1 \paren {a u}^m \paren {a^n - \paren {a u}^n}^p \rd u
| c = substituting $x = a u$
}}
{{eqn | r = a \times a^m \times a^{n p} \int_0^1 u^m \paren {1 - u^n}^p \rd u
}}
{{eqn | r = \frac {a^{m + 1 + n p} } n \int_0^1 \frac {u^m}... | :$\ds \int_0^a x^m \paren {a^n - x^n}^p \rd x = \frac {a^{m + 1 + n p} \, \map \Gamma {\frac {m + 1} n} \map \Gamma {p + 1} } {n \map \Gamma {\frac {m + 1} n + p + 1} }$ | {{begin-eqn}}
{{eqn | l = \int_0^a x^m \paren {a^n - x^n}^p \rd x
| r = a \int_0^1 \paren {a u}^m \paren {a^n - \paren {a u}^n}^p \rd u
| c = [[Integration by Substitution|substituting]] $x = a u$
}}
{{eqn | r = a \times a^m \times a^{n p} \int_0^1 u^m \paren {1 - u^n}^p \rd u
}}
{{eqn | r = \frac {a^{m + 1... | Definite Integral from 0 to a of x^m by (a^n - x^n)^p | https://proofwiki.org/wiki/Definite_Integral_from_0_to_a_of_x^m_by_(a^n_-_x^n)^p | https://proofwiki.org/wiki/Definite_Integral_from_0_to_a_of_x^m_by_(a^n_-_x^n)^p | [
"Examples of Definite Integrals"
] | [] | [
"Integration by Substitution",
"Integration by Substitution"
] |
proofwiki-14205 | Equivalence of Definitions of Complement of Subgroup | Let $G$ be a group with identity $e$.
Let $H$ and $K$ be subgroups.
{{TFAE|def = Complement of Subgroup}} | === Definition $1$ implies Definition $2$ ===
Let $G = H K$.
Then $H K$ is a group.
By Subset Product of Subgroups:
:$H K = K H$
Thus $K H = G$.
{{qed|lemma}} | Let $G$ be a [[Definition:Group|group]] with [[Definition:Identity Element|identity]] $e$.
Let $H$ and $K$ be [[Definition:Subgroup|subgroups]].
{{TFAE|def = Complement of Subgroup}} | === Definition $1$ implies Definition $2$ ===
Let $G = H K$.
Then $H K$ is a [[Definition:Group|group]].
By [[Subset Product of Subgroups]]:
:$H K = K H$
Thus $K H = G$.
{{qed|lemma}} | Equivalence of Definitions of Complement of Subgroup | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complement_of_Subgroup | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Complement_of_Subgroup | [
"Subgroup Complements"
] | [
"Definition:Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Subgroup"
] | [
"Definition:Group",
"Subset Product of Subgroups",
"Definition:Group",
"Subset Product of Subgroups"
] |
proofwiki-14206 | Equivalent Characterizations of Finer Equivalence Relation | Let $X$ be a set.
Let $\equiv$ and $\sim$ be equivalence relations on $X$.
{{TFAE}}
# $\equiv$ is finer than $\sim$:
#:$\forall x, y \in X : x \equiv y \implies x \sim y$
#The graph of $\equiv$ is contained in the graph of $\sim$.
#Every $\equiv$-equivalence class is contained in a $\sim$-equivalence class.
#Every $\si... | === 1 implies 2 ===
{{begin-eqn}}
{{eqn | l = \tuple {x, y}
| o = \in
| r = \map \TT \equiv
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \equiv
| r = y
| c = {{Defof|Graph of Relation}}
}}
{{eqn | ll= \leadsto
| l = x
| o = \sim
| r = y
| c = from $1$
}}
{{eqn... | Let $X$ be a [[Definition:Set|set]].
Let $\equiv$ and $\sim$ be [[Definition:equivalence Relation|equivalence relations]] on $X$.
{{TFAE}}
# $\equiv$ is [[Definition:Finer Equivalence Relation|finer]] than $\sim$:
#:$\forall x, y \in X : x \equiv y \implies x \sim y$
#The [[Definition:Graph of Relation|graph]] of $\... | === 1 implies 2 ===
{{begin-eqn}}
{{eqn | l = \tuple {x, y}
| o = \in
| r = \map \TT \equiv
}}
{{eqn | ll= \leadstoandfrom
| l = x
| o = \equiv
| r = y
| c = {{Defof|Graph of Relation}}
}}
{{eqn | ll= \leadsto
| l = x
| o = \sim
| r = y
| c = from $1$
}}
{{eq... | Equivalent Characterizations of Finer Equivalence Relation | https://proofwiki.org/wiki/Equivalent_Characterizations_of_Finer_Equivalence_Relation | https://proofwiki.org/wiki/Equivalent_Characterizations_of_Finer_Equivalence_Relation | [
"Equivalence Relations"
] | [
"Definition:Set",
"Definition:equivalence Relation",
"Definition:Finer Equivalence Relation",
"Definition:Relation/Graph",
"Definition:Subset",
"Definition:Relation/Graph",
"Definition:Equivalence Class",
"Definition:Subset",
"Definition:Equivalence Class",
"Definition:Equivalence Class",
"Defin... | [
"Definition:Subset",
"Definition:Subset"
] |
proofwiki-14207 | Equivalence of Definitions of Finer Topology | Let $S$ be a set.
Let $\tau_1$ and $\tau_2$ be topologies on $S$.
{{TFAE|def = Finer Topology}} | Let $I_S: \struct {S, \tau_1} \to \struct {S, \tau_2}$ be the identity mapping on $S$.
Then:
{{begin-eqn}}
{{eqn | l = \tau_1
| o = \supseteq
| r = \tau_2
| c = {{Defof|Finer Topology|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| q = \forall U \subseteq S
| l = U \in \tau_2
| o = \imp... | Let $S$ be a [[Definition:Set|set]].
Let $\tau_1$ and $\tau_2$ be [[Definition:Topology|topologies]] on $S$.
{{TFAE|def = Finer Topology}} | Let $I_S: \struct {S, \tau_1} \to \struct {S, \tau_2}$ be the [[Definition:Identity Mapping|identity mapping]] on $S$.
Then:
{{begin-eqn}}
{{eqn | l = \tau_1
| o = \supseteq
| r = \tau_2
| c = {{Defof|Finer Topology|index = 1}}
}}
{{eqn | ll= \leadstoandfrom
| q = \forall U \subseteq S
|... | Equivalence of Definitions of Finer Topology | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Finer_Topology | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Finer_Topology | [
"Topology"
] | [
"Definition:Set",
"Definition:Topology"
] | [
"Definition:Identity Mapping",
"Definition:Continuous Mapping (Topology)",
"Definition:Continuous Mapping (Topology)",
"Category:Topology"
] |
proofwiki-14208 | Definite Integral from 0 to Infinity of x^m over (x^n + a^n)^r | :$\ds \int_0^\infty \frac {x^m \rd x} {\paren {a^n + x^n}^r} = \frac {\paren {-1}^{r - 1} \pi a^{m + 1 - n r} \, \map \Gamma {\frac {m + 1} n} } {n \sin \frac {\paren {m + 1} \pi} n \paren {r - 1}! \, \map \Gamma {\frac {m + 1} n - r + 1} }$
for:
:$0 < m + 1 < n r$
:$r + 1 - \dfrac {m + 1} n \notin \N_{>0}$
:$r \in \N_... | {{begin-eqn}}
{{eqn | l = u
| r = \frac {x^n} {a^n}
| c =
}}
{{eqn | ll= \leadsto
| l = \d u
| r = \frac {n x^{n - 1} } {a^n} \rd x
| c = Power Rule for Derivatives
}}
{{eqn | l = \d u
| r = \frac {n \paren {a \times u^{\frac 1 n} }^{n - 1} } {a^n} \rd x
| c = substituting for ... | :$\ds \int_0^\infty \frac {x^m \rd x} {\paren {a^n + x^n}^r} = \frac {\paren {-1}^{r - 1} \pi a^{m + 1 - n r} \, \map \Gamma {\frac {m + 1} n} } {n \sin \frac {\paren {m + 1} \pi} n \paren {r - 1}! \, \map \Gamma {\frac {m + 1} n - r + 1} }$
for:
:$0 < m + 1 < n r$
:$r + 1 - \dfrac {m + 1} n \notin \N_{>0}$
:$r \in \N... | {{begin-eqn}}
{{eqn | l = u
| r = \frac {x^n} {a^n}
| c =
}}
{{eqn | ll= \leadsto
| l = \d u
| r = \frac {n x^{n - 1} } {a^n} \rd x
| c = [[Power Rule for Derivatives]]
}}
{{eqn | l = \d u
| r = \frac {n \paren {a \times u^{\frac 1 n} }^{n - 1} } {a^n} \rd x
| c = substituting ... | Definite Integral from 0 to Infinity of x^m over (x^n + a^n)^r | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Infinity_of_x^m_over_(x^n_+_a^n)^r | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Infinity_of_x^m_over_(x^n_+_a^n)^r | [
"Examples of Definite Integrals"
] | [] | [
"Power Rule for Derivatives",
"Integration by Substitution",
"Beta Function as Integral of Power of t over Power of t plus 1",
"Euler's Reflection Formula",
"Sine of Angle plus Integer Multiple of Pi",
"Gamma Function Extends Factorial"
] |
proofwiki-14209 | Definite Integral from 0 to Pi of Sine of m x by Sine of n x | Let $m, n \in \Z$ be integers.
Then:
:$\ds \int_0^\pi \sin m x \sin n x \rd x = \begin{cases} 0 & : m \ne n \\ \dfrac \pi 2 & : m = n \end{cases}$
That is:
:$\ds \int_0^\pi \sin m x \sin n x \rd x = \dfrac \pi 2 \delta_{m n}$
where $\delta_{m n}$ is the Kronecker delta. | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \sin m x \sin n x \rd x
| r = \frac {\sin \paren {m - n} x} {2 \paren {m - n} } - \frac {\sin \paren {m + n} x} {2 \paren {m + n} } + C
| c = Primitive of $\sin m x \sin n x$
}}
{{eqn | ll= \leadsto
| l = \int_0^\pi \sin m x \sin n x \rd x
| r = \int... | Let $m, n \in \Z$ be [[Definition:Integer|integers]].
Then:
:$\ds \int_0^\pi \sin m x \sin n x \rd x = \begin{cases} 0 & : m \ne n \\ \dfrac \pi 2 & : m = n \end{cases}$
That is:
:$\ds \int_0^\pi \sin m x \sin n x \rd x = \dfrac \pi 2 \delta_{m n}$
where $\delta_{m n}$ is the [[Definition:Kronecker Delta|Kronecke... | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \sin m x \sin n x \rd x
| r = \frac {\sin \paren {m - n} x} {2 \paren {m - n} } - \frac {\sin \paren {m + n} x} {2 \paren {m + n} } + C
| c = [[Primitive of Sine of a x by Sine of b x|Primitive of $\sin m x \sin n x$]]
}}
{{eqn | ll= \leadsto
| l = \int_0... | Definite Integral from 0 to Pi of Sine of m x by Sine of n x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_Sine_of_m_x_by_Sine_of_n_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_Sine_of_m_x_by_Sine_of_n_x | [
"Definite Integrals involving Sine Function"
] | [
"Definition:Integer",
"Definition:Kronecker Delta"
] | [
"Primitive of Sine of a x by Sine of b x",
"Sine of Zero is Zero",
"Sine of Integer Multiple of Pi",
"Primitive of Square of Sine of a x",
"Sine of Zero is Zero",
"Sine of Integer Multiple of Pi"
] |
proofwiki-14210 | Definite Integral from 0 to Pi of Cosine of m x by Cosine of n x | Let $m, n \in \Z$ be integers.
Then:
:$\ds \int_0^\pi \cos m x \cos n x \rd x = \begin {cases} 0 & : m \ne n \\ \dfrac \pi 2 & : m = n \end {cases}$
That is:
:$\ds \int_0^\pi \cos m x \cos n x \rd x = \dfrac \pi 2 \delta_{m n}$
where $\delta_{m n}$ is the Kronecker delta. | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \cos m x \cos n x \rd x
| r = \frac {\map \sin {\paren {m - n} x} } {2 \paren {m - n} } + \frac {\map \sin {\paren {m + n} x} } {2 \paren {m + n} } + C
| c = Primitive of $\cos m x \cos n x$
}}
{{eqn | ll= \leadsto
| l = \int_0^\pi \cos m x \cos n x \rd x
... | Let $m, n \in \Z$ be [[Definition:Integer|integers]].
Then:
:$\ds \int_0^\pi \cos m x \cos n x \rd x = \begin {cases} 0 & : m \ne n \\ \dfrac \pi 2 & : m = n \end {cases}$
That is:
:$\ds \int_0^\pi \cos m x \cos n x \rd x = \dfrac \pi 2 \delta_{m n}$
where $\delta_{m n}$ is the [[Definition:Kronecker Delta|Kroneck... | Let $m \ne n$.
{{begin-eqn}}
{{eqn | l = \int \cos m x \cos n x \rd x
| r = \frac {\map \sin {\paren {m - n} x} } {2 \paren {m - n} } + \frac {\map \sin {\paren {m + n} x} } {2 \paren {m + n} } + C
| c = [[Primitive of Cosine of a x by Cosine of b x|Primitive of $\cos m x \cos n x$]]
}}
{{eqn | ll= \leadst... | Definite Integral from 0 to Pi of Cosine of m x by Cosine of n x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_Cosine_of_m_x_by_Cosine_of_n_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_Cosine_of_m_x_by_Cosine_of_n_x | [
"Definite Integrals involving Cosine Function"
] | [
"Definition:Integer",
"Definition:Kronecker Delta"
] | [
"Primitive of Cosine of a x by Cosine of b x",
"Sine of Zero is Zero",
"Sine of Integer Multiple of Pi",
"Primitive of Square of Cosine of a x",
"Sine of Zero is Zero",
"Sine of Integer Multiple of Pi"
] |
proofwiki-14211 | Definite Integral from 0 to Pi of Sine of m x by Cosine of n x | Let $m, n \in \Z$ be integers.
Then:
:$\ds \int_0^\pi \sin m x \cos n x \rd x = \begin{cases} 0 & : m + n \text { even} \\ \dfrac {2 m} {m^2 - n^2} & : m + n \text { odd} \end{cases}$ | First we address the special case where $m = n$.
In this case $m + n = m + m = 2 m$ is even.
We have:
{{begin-eqn}}
{{eqn | l = \int \sin m x \cos m x \rd x
| r = \frac {\sin^2 m x} {2 m} + C
| c = Primitive of $\sin m x \cos m x$
}}
{{eqn | ll= \leadsto
| l = \int_0^\pi \sin m x \cos m x \rd x
... | Let $m, n \in \Z$ be [[Definition:Integer|integers]].
Then:
:$\ds \int_0^\pi \sin m x \cos n x \rd x = \begin{cases} 0 & : m + n \text { even} \\ \dfrac {2 m} {m^2 - n^2} & : m + n \text { odd} \end{cases}$ | First we address the special case where $m = n$.
In this case $m + n = m + m = 2 m$ is [[Definition:Even Integer|even]].
We have:
{{begin-eqn}}
{{eqn | l = \int \sin m x \cos m x \rd x
| r = \frac {\sin^2 m x} {2 m} + C
| c = [[Primitive of Sine of a x by Cosine of a x|Primitive of $\sin m x \cos m x$]]... | Definite Integral from 0 to Pi of Sine of m x by Cosine of n x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_Sine_of_m_x_by_Cosine_of_n_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Pi_of_Sine_of_m_x_by_Cosine_of_n_x | [
"Definite Integrals involving Cosine Function",
"Definite Integrals involving Sine Function"
] | [
"Definition:Integer"
] | [
"Definition:Even Integer",
"Primitive of Sine of a x by Cosine of a x",
"Sine of Integer Multiple of Pi",
"Definition:Even Integer",
"Primitive of Sine of a x by Cosine of b x",
"Cosine of Zero is One",
"Cosine of Integer Multiple of Pi",
"Definition:Parity of Integer",
"Definition:Even Integer",
... |
proofwiki-14212 | Definite Integral from 0 to Half Pi of Square of Sine x | :$\ds \int_0^{\frac \pi 2} \sin^2 x \rd x = \frac \pi 4$ | {{begin-eqn}}
{{eqn | l = \int \sin^2 x \rd x
| r = \frac x 2 - \frac {\sin 2 x} 4 + C
| c = Primitive of $\sin^2 x$
}}
{{eqn | ll= \leadsto
| l = \int_0^{\frac \pi 2} \sin^2 x \rd x
| r = \intlimits {\frac x 2 - \frac {\sin 2 x} 4} 0 {\frac \pi 2}
| c =
}}
{{eqn | r = \paren {\frac \pi 4... | :$\ds \int_0^{\frac \pi 2} \sin^2 x \rd x = \frac \pi 4$ | {{begin-eqn}}
{{eqn | l = \int \sin^2 x \rd x
| r = \frac x 2 - \frac {\sin 2 x} 4 + C
| c = [[Primitive of Square of Sine of a x|Primitive of $\sin^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int_0^{\frac \pi 2} \sin^2 x \rd x
| r = \intlimits {\frac x 2 - \frac {\sin 2 x} 4} 0 {\frac \pi 2}
| ... | Definite Integral from 0 to Half Pi of Square of Sine x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Sine_x/Proof_1 | [
"Definite Integrals involving Sine Function",
"Definite Integral from 0 to Half Pi of Square of Sine x"
] | [] | [
"Primitive of Square of Sine of a x",
"Sine of Integer Multiple of Pi"
] |
proofwiki-14213 | Definite Integral from 0 to Half Pi of Square of Sine x | :$\ds \int_0^{\frac \pi 2} \sin^2 x \rd x = \frac \pi 4$ | We have:
{{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \sin^2 x \rd x
| r = \int_0^{\frac \pi 2} \sin^2 \paren {\frac \pi 2 - x} \rd x
| c = Integral between Limits is Independent of Direction
}}
{{eqn | r = \int_0^{\frac \pi 2} \cos^2 x \rd x
| c = Sine of Complement equals Cosine
}}
{{end-eqn}}
So:
{{begin-eqn}}... | :$\ds \int_0^{\frac \pi 2} \sin^2 x \rd x = \frac \pi 4$ | We have:
{{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \sin^2 x \rd x
| r = \int_0^{\frac \pi 2} \sin^2 \paren {\frac \pi 2 - x} \rd x
| c = [[Integral between Limits is Independent of Direction]]
}}
{{eqn | r = \int_0^{\frac \pi 2} \cos^2 x \rd x
| c = [[Sine of Complement equals Cosine]]
}}
{{end-eqn}}
So:
{{... | Definite Integral from 0 to Half Pi of Square of Sine x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Sine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Sine_x/Proof_2 | [
"Definite Integrals involving Sine Function",
"Definite Integral from 0 to Half Pi of Square of Sine x"
] | [] | [
"Integral between Limits is Independent of Direction",
"Sine of Complement equals Cosine",
"Sum of Squares of Sine and Cosine",
"Primitive of Constant"
] |
proofwiki-14214 | Definite Integral from 0 to Half Pi of Square of Cosine x | :$\ds \int_0^{\frac \pi 2} \cos^2 x \rd x = \frac \pi 4$ | {{begin-eqn}}
{{eqn | l = \int \cos^2 x \rd x
| r = \frac x 2 + \frac {\sin 2 x} 4 + C
| c = Primitive of $\cos^2 x$
}}
{{eqn | ll= \leadsto
| l = \int_0^{\frac \pi 2} \cos^2 x \rd x
| r = \intlimits {\frac x 2 + \frac {\sin 2 x} 4} 0 {\frac \pi 2}
| c =
}}
{{eqn | r = \paren {\frac \pi 4... | :$\ds \int_0^{\frac \pi 2} \cos^2 x \rd x = \frac \pi 4$ | {{begin-eqn}}
{{eqn | l = \int \cos^2 x \rd x
| r = \frac x 2 + \frac {\sin 2 x} 4 + C
| c = [[Primitive of Square of Cosine of a x|Primitive of $\cos^2 x$]]
}}
{{eqn | ll= \leadsto
| l = \int_0^{\frac \pi 2} \cos^2 x \rd x
| r = \intlimits {\frac x 2 + \frac {\sin 2 x} 4} 0 {\frac \pi 2}
... | Definite Integral from 0 to Half Pi of Square of Cosine x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Cosine_x/Proof_1 | [
"Definite Integrals involving Cosine Function",
"Definite Integral from 0 to Half Pi of Square of Cosine x"
] | [] | [
"Primitive of Square of Cosine of a x",
"Sine of Integer Multiple of Pi"
] |
proofwiki-14215 | Definite Integral from 0 to Half Pi of Square of Cosine x | :$\ds \int_0^{\frac \pi 2} \cos^2 x \rd x = \frac \pi 4$ | We have:
{{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \cos^2 x \rd x
| r = \int_0^{\frac \pi 2} \map {\cos^2} {\frac \pi 2 - x} \rd x
| c = Integral between Limits is Independent of Direction
}}
{{eqn | r = \int_0^{\frac \pi 2} \sin^2 x \rd x
| c = Cosine of Complement equals Sine
}}
{{end-eqn}}
So:
{{begin-eqn}}... | :$\ds \int_0^{\frac \pi 2} \cos^2 x \rd x = \frac \pi 4$ | We have:
{{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \cos^2 x \rd x
| r = \int_0^{\frac \pi 2} \map {\cos^2} {\frac \pi 2 - x} \rd x
| c = [[Integral between Limits is Independent of Direction]]
}}
{{eqn | r = \int_0^{\frac \pi 2} \sin^2 x \rd x
| c = [[Cosine of Complement equals Sine]]
}}
{{end-eqn}}
So:
{{... | Definite Integral from 0 to Half Pi of Square of Cosine x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Square_of_Cosine_x/Proof_2 | [
"Definite Integrals involving Cosine Function",
"Definite Integral from 0 to Half Pi of Square of Cosine x"
] | [] | [
"Integral between Limits is Independent of Direction",
"Cosine of Complement equals Sine",
"Sum of Squares of Sine and Cosine",
"Primitive of Constant"
] |
proofwiki-14216 | Reduction Formula for Definite Integral of Power of Cosine | Let $n \in \Z_{> 0}$ be a positive integer.
Let $I_n$ be defined as:
:$\ds I_n = \int_0^{\frac \pi 2} \cos^n x \rd x$
Then $\sequence {I_n}$ is a decreasing sequence of real numbers which satisfies:
:$n I_n = \paren {n - 1} I_{n - 2}$
Thus:
:$I_n = \dfrac {n - 1} n I_{n - 2}$
is a reduction formula for $I_n$. | From Shape of Cosine Function:
:$\forall x \in \closedint 0 {\dfrac \pi 2}: 0 \le \cos x \le 1$
So, on the same interval:
:$0 \le \cos^{n + 1} x \le \cos^n x$
Therefore:
:$\forall n \in \N: 0 < I_{n + 1} < I_n$
From Reduction Formula for Integral of Power of Cosine:
:$\ds \int \cos^n x \rd x = \dfrac {\cos^{n - 1} x \s... | Let $n \in \Z_{> 0}$ be a [[Definition:Positive Integer|positive integer]].
Let $I_n$ be defined as:
:$\ds I_n = \int_0^{\frac \pi 2} \cos^n x \rd x$
Then $\sequence {I_n}$ is a [[Definition:Decreasing Real Sequence|decreasing sequence of real numbers]] which satisfies:
:$n I_n = \paren {n - 1} I_{n - 2}$
Thus:
:$... | From [[Shape of Cosine Function]]:
:$\forall x \in \closedint 0 {\dfrac \pi 2}: 0 \le \cos x \le 1$
So, on the same [[Definition:Open Real Interval|interval]]:
:$0 \le \cos^{n + 1} x \le \cos^n x$
Therefore:
:$\forall n \in \N: 0 < I_{n + 1} < I_n$
From [[Reduction Formula for Integral of Power of Cosine]]:
:$\ds ... | Reduction Formula for Definite Integral of Power of Cosine | https://proofwiki.org/wiki/Reduction_Formula_for_Definite_Integral_of_Power_of_Cosine | https://proofwiki.org/wiki/Reduction_Formula_for_Definite_Integral_of_Power_of_Cosine | [
"Reduction Formula for Definite Integral of Power of Cosine",
"Reduction Formulae (Calculus)",
"Definite Integrals involving Cosine Function"
] | [
"Definition:Positive/Integer",
"Definition:Decreasing/Sequence/Real Sequence",
"Definition:Reduction Formula (Calculus)"
] | [
"Shape of Cosine Function",
"Definition:Real Interval/Open",
"Reduction Formula for Integral of Power of Cosine",
"Cosine of Right Angle",
"Sine of Zero is Zero"
] |
proofwiki-14217 | Definite Integral from 0 to Half Pi of Even Power of Cosine x | :$\ds \int_0^{\frac \pi 2} \cos^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$ | The proof proceeds by induction.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$\ds \int_0^{\frac \pi 2} \cos^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$
=== Basis for the Induction ===
$\map P 1$ is the case:
{{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \cos^2 x \rd x
... | :$\ds \int_0^{\frac \pi 2} \cos^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \int_0^{\frac \pi 2} \cos^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$
=== Basis for the Induction ===
$\map P 1$ is the ... | Definite Integral from 0 to Half Pi of Even Power of Cosine x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Even_Power_of_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Even_Power_of_Cosine_x/Proof_1 | [
"Definite Integral from 0 to Half Pi of Even Power of Cosine x",
"Reduction Formula for Integral of Power of Cosine",
"Definite Integrals involving Cosine Function"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definite Integral from 0 to Half Pi of Square of Cosine x",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Reduction Formula for Definite Integral of Power of Cosine",
"Definite In... |
proofwiki-14218 | Definite Integral from 0 to Half Pi of Even Power of Cosine x | :$\ds \int_0^{\frac \pi 2} \cos^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$ | {{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \cos^{2 n} x \rd x
| r = \int_0^{\frac \pi 2} \paren {\sin x}^{\frac 2 2 - 1} \paren {\cos x}^{2 \paren {n + \frac 1 2} - 1} \rd x
}}
{{eqn | r = \frac 1 2 \Beta \paren {\frac 1 2, n + \frac 1 2}
| c = {{Defof|Beta Function|index = 2}}
}}
{{eqn | r = \frac 1 2 \cdot \fra... | :$\ds \int_0^{\frac \pi 2} \cos^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$ | {{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \cos^{2 n} x \rd x
| r = \int_0^{\frac \pi 2} \paren {\sin x}^{\frac 2 2 - 1} \paren {\cos x}^{2 \paren {n + \frac 1 2} - 1} \rd x
}}
{{eqn | r = \frac 1 2 \Beta \paren {\frac 1 2, n + \frac 1 2}
| c = {{Defof|Beta Function|index = 2}}
}}
{{eqn | r = \frac 1 2 \cdot \fra... | Definite Integral from 0 to Half Pi of Even Power of Cosine x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Even_Power_of_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Even_Power_of_Cosine_x/Proof_2 | [
"Definite Integral from 0 to Half Pi of Even Power of Cosine x",
"Reduction Formula for Integral of Power of Cosine",
"Definite Integrals involving Cosine Function"
] | [] | [
"Gamma Function of One Half",
"Gamma Function of Positive Half-Integer"
] |
proofwiki-14219 | Definite Integral from 0 to Half Pi of Odd Power of Cosine x | :$\ds \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$ | The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
:$\ds \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$
=== Basis for the Induction ===
$\map P 0$ is the case:
{{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \cos x \rd x
| ... | :$\ds \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$ | 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]]:
:$\ds \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$
=== Basis for the Induction ===
$\map P 0$ is the cas... | Definite Integral from 0 to Half Pi of Odd Power of Cosine x/Proof 1 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Odd_Power_of_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Odd_Power_of_Cosine_x/Proof_1 | [
"Definite Integral from 0 to Half Pi of Odd Power of Cosine x",
"Reduction Formula for Integral of Power of Cosine",
"Definite Integrals involving Cosine Function"
] | [] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Primitive of Cosine Function/Corollary",
"Sine of Right Angle",
"Sine of Zero is Zero",
"Definition:Basis for the Induction",
"Definition:Induction Hypothesis",
"Definition:Induction Step",
"Reduction Formula for Definite Integral of ... |
proofwiki-14220 | Definite Integral from 0 to Half Pi of Odd Power of Cosine x | :$\ds \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$ | {{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x
| r = \int_0^{\frac \pi 2} \paren {\sin x}^{\frac 2 2 - 1} \paren {\cos x}^{2 \paren {n + 1} - 1} \rd x
}}
{{eqn | r = \frac 1 2 \map \Beta {\frac 1 2, n + 1}
| c = {{Defof|Beta Function|index = 2}}
}}
{{eqn | r = \frac 1 2 \cdot \frac {\... | :$\ds \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x = \dfrac {\paren {2^n n!}^2} {\paren {2 n + 1}!}$ | {{begin-eqn}}
{{eqn | l = \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x
| r = \int_0^{\frac \pi 2} \paren {\sin x}^{\frac 2 2 - 1} \paren {\cos x}^{2 \paren {n + 1} - 1} \rd x
}}
{{eqn | r = \frac 1 2 \map \Beta {\frac 1 2, n + 1}
| c = {{Defof|Beta Function|index = 2}}
}}
{{eqn | r = \frac 1 2 \cdot \frac {\... | Definite Integral from 0 to Half Pi of Odd Power of Cosine x/Proof 2 | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Odd_Power_of_Cosine_x | https://proofwiki.org/wiki/Definite_Integral_from_0_to_Half_Pi_of_Odd_Power_of_Cosine_x/Proof_2 | [
"Definite Integral from 0 to Half Pi of Odd Power of Cosine x",
"Reduction Formula for Integral of Power of Cosine",
"Definite Integrals involving Cosine Function"
] | [] | [
"Gamma Function Extends Factorial",
"Gamma Function of One Half",
"Gamma Function of Positive Half-Integer"
] |
proofwiki-14221 | Arctangent of Zero is Zero | :$\arctan 0 = 0$ | By definition, $\arctan$ is the inverse of the tangent function's restriction to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
By Tangent of Zero:
:$\tan 0 = 0$
As $0 \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$, we have $\arctan 0 = 0$ by the definition of an inverse function.
{{qed}}
Category:Arctangent Function
c9xupu... | :$\arctan 0 = 0$ | By [[Definition:Real Arctangent|definition]], $\arctan$ is the [[Definition:Inverse of Mapping|inverse]] of the [[Definition:Real Tangent Function|tangent function]]'s [[Definition:Restriction of Mapping|restriction]] to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
By [[Tangent of Zero]]:
:$\tan 0 = 0$
As $0 \in \open... | Arctangent of Zero is Zero | https://proofwiki.org/wiki/Arctangent_of_Zero_is_Zero | https://proofwiki.org/wiki/Arctangent_of_Zero_is_Zero | [
"Arctangent Function"
] | [] | [
"Definition:Inverse Tangent/Real/Arctangent",
"Definition:Inverse of Mapping",
"Definition:Tangent Function/Real",
"Definition:Restriction/Mapping",
"Tangent of Zero",
"Definition:Inverse of Mapping",
"Category:Arctangent Function"
] |
proofwiki-14222 | Integral to Infinity of Sine p x Cosine q x over x | :$\ds \int_0^\infty \frac {\sin p x \cos q x} x \rd x = \begin {cases} \dfrac \pi 2 & : p > q > 0 \\ \\ 0 & : 0 < p < q \\ \\ \dfrac \pi 4 & : p = q > 0 \end {cases}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sin p x \cos q x} x \rd x
| r = \int_0^\infty \frac 1 2 \cdot \frac {\sin \paren {\paren {p + q} x} + \sin \paren {\paren {p - q} x} } x \rd x
| c = Werner Formula for Sine by Cosine
}}
{{eqn | r = \frac 1 2 \int_0^\infty \frac {\sin \paren {\paren {p + q} x} ... | :$\ds \int_0^\infty \frac {\sin p x \cos q x} x \rd x = \begin {cases} \dfrac \pi 2 & : p > q > 0 \\ \\ 0 & : 0 < p < q \\ \\ \dfrac \pi 4 & : p = q > 0 \end {cases}$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\sin p x \cos q x} x \rd x
| r = \int_0^\infty \frac 1 2 \cdot \frac {\sin \paren {\paren {p + q} x} + \sin \paren {\paren {p - q} x} } x \rd x
| c = [[Werner Formula for Sine by Cosine]]
}}
{{eqn | r = \frac 1 2 \int_0^\infty \frac {\sin \paren {\paren {p + q}... | Integral to Infinity of Sine p x Cosine q x over x | https://proofwiki.org/wiki/Integral_to_Infinity_of_Sine_p_x_Cosine_q_x_over_x | https://proofwiki.org/wiki/Integral_to_Infinity_of_Sine_p_x_Cosine_q_x_over_x | [
"Definite Integrals involving Sine Function",
"Definite Integrals involving Cosine Function"
] | [] | [
"Werner Formulas/Sine by Cosine",
"Linear Combination of Integrals/Definite",
"Integration by Substitution",
"Integration by Substitution",
"Definition:Strictly Positive/Real Number",
"Dirichlet Integral",
"Definition:Strictly Positive/Real Number",
"Dirichlet Integral",
"Integration by Substitution... |
proofwiki-14223 | Integral to Infinity of Sine p x Sine q x over x Squared | :$\ds \int_0^\infty \frac {\sin p x \sin q x} {x^2} \rd x = \begin {cases} \dfrac {\pi p} 2 & : 0 < p \le q \\ \\ \dfrac {\pi q} 2 & : p \ge q > 0 \end {cases}$ | With a view to expressing the primitive in the form:
:$\ds \int f g' \rd t = f g - \int f' g \rd t$
let:
{{begin-eqn}}
{{eqn | l = f
| r = \sin p x \sin q x
}}
{{eqn | ll= \leadsto
| l = f'
| r = p \cos p x \sin q x + q \sin p x \cos q x
| c = Product Rule for Derivatives and Derivative of $\sin... | :$\ds \int_0^\infty \frac {\sin p x \sin q x} {x^2} \rd x = \begin {cases} \dfrac {\pi p} 2 & : 0 < p \le q \\ \\ \dfrac {\pi q} 2 & : p \ge q > 0 \end {cases}$ | With a view to expressing the [[Definition:Primitive (Calculus)|primitive]] in the form:
:$\ds \int f g' \rd t = f g - \int f' g \rd t$
let:
{{begin-eqn}}
{{eqn | l = f
| r = \sin p x \sin q x
}}
{{eqn | ll= \leadsto
| l = f'
| r = p \cos p x \sin q x + q \sin p x \cos q x
| c = [[Product Rule... | Integral to Infinity of Sine p x Sine q x over x Squared | https://proofwiki.org/wiki/Integral_to_Infinity_of_Sine_p_x_Sine_q_x_over_x_Squared | https://proofwiki.org/wiki/Integral_to_Infinity_of_Sine_p_x_Sine_q_x_over_x_Squared | [
"Definite Integrals involving Sine Function"
] | [] | [
"Definition:Primitive (Calculus)",
"Product Rule for Derivatives",
"Derivative of Sine Function/Corollary",
"Derivative of Composite Function",
"Integration by Parts",
"L'Hôpital's Rule",
"Combination Theorem for Limits of Functions/Product Rule",
"Linear Combination of Integrals/Definite"
] |
proofwiki-14224 | Order of Finite p-Group is Power of p | Let $G$ be a finite group.
Let $p$ be a prime number.
Let all elements of $G$ have order a power of $p$.
Then $G$ is a $p$-group. | {{AimForCont}}:
:$\order G = k p^n: p \nmid k$
where $\order G$ denotes the order of $G$.
By Divisors of Power of Prime:
:$k \nmid p^n$
From the First Sylow Theorem:
:$\exists H \le G: \order H = k$
where $H \le G$ denotes that $H$ is a subgroup of $G$.
Thus:
:$\exists h \in H: \order h \divides k \implies \order h \nm... | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $p$ be a [[Definition:Prime Number|prime number]].
Let all [[Definition:Element|elements]] of $G$ have [[Definition:Order of Group Element|order]] a [[Definition:Prime Power|power of $p$]].
Then $G$ is a [[Definition:P-Group|$p$-group]]. | {{AimForCont}}:
:$\order G = k p^n: p \nmid k$
where $\order G$ denotes the [[Definition:Order of Structure|order]] of $G$.
By [[Divisors of Power of Prime]]:
:$k \nmid p^n$
From the [[First Sylow Theorem]]:
:$\exists H \le G: \order H = k$
where $H \le G$ denotes that $H$ is a [[Definition:Subgroup|subgroup]] of ... | Order of Finite p-Group is Power of p/Proof 1 | https://proofwiki.org/wiki/Order_of_Finite_p-Group_is_Power_of_p | https://proofwiki.org/wiki/Order_of_Finite_p-Group_is_Power_of_p/Proof_1 | [
"Order of Finite p-Group is Power of p",
"P-Groups",
"Finite Groups"
] | [
"Definition:Finite Group",
"Definition:Prime Number",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Prime Power",
"Definition:P-Group"
] | [
"Definition:Order of Structure",
"Divisors of Power of Prime",
"First Sylow Theorem",
"Definition:Subgroup",
"Definition:Divisor (Algebra)/Integer",
"Proof by Contradiction",
"Definition:Prime Power"
] |
proofwiki-14225 | Order of Finite p-Group is Power of p | Let $G$ be a finite group.
Let $p$ be a prime number.
Let all elements of $G$ have order a power of $p$.
Then $G$ is a $p$-group. | Let every element of $G$ be a $p$-element.
Let $q$ be a prime number which is a divisor of the order $\order G$ of $G$.
By Cauchy's Lemma (Group Theory), there exists an element of $G$ whose order is a divisor of $q$.
But as the order of all elements of $G$ divide $p^n$ it follows that $q = p$.
Thus $G$ is a group whos... | Let $G$ be a [[Definition:Finite Group|finite group]].
Let $p$ be a [[Definition:Prime Number|prime number]].
Let all [[Definition:Element|elements]] of $G$ have [[Definition:Order of Group Element|order]] a [[Definition:Prime Power|power of $p$]].
Then $G$ is a [[Definition:P-Group|$p$-group]]. | Let every [[Definition:Element|element]] of $G$ be a [[Definition:P-Element|$p$-element]].
Let $q$ be a [[Definition:Prime Number|prime number]] which is a [[Definition:Divisor of Integer|divisor]] of the [[Definition:Order of Structure|order]] $\order G$ of $G$.
By [[Cauchy's Lemma (Group Theory)]], there exists an ... | Order of Finite p-Group is Power of p/Proof 2 | https://proofwiki.org/wiki/Order_of_Finite_p-Group_is_Power_of_p | https://proofwiki.org/wiki/Order_of_Finite_p-Group_is_Power_of_p/Proof_2 | [
"Order of Finite p-Group is Power of p",
"P-Groups",
"Finite Groups"
] | [
"Definition:Finite Group",
"Definition:Prime Number",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Prime Power",
"Definition:P-Group"
] | [
"Definition:Element",
"Definition:P-Element",
"Definition:Prime Number",
"Definition:Divisor (Algebra)/Integer",
"Definition:Order of Structure",
"Cauchy's Lemma (Group Theory)",
"Definition:Element",
"Definition:Order of Group Element",
"Definition:Divisor (Algebra)/Integer",
"Definition:Order of... |
proofwiki-14226 | Sum of Sequence of Binomial Coefficients by Sum of Powers of Integers | Let $n, k \in \Z_{\ge 0}$ be positive integers.
Let $S_k = \ds \sum_{i \mathop = 1}^n i^k$.
Then:
:$\ds \sum_{i \mathop = 1}^k \binom {k + 1} i S_i = \paren {n + 1}^{k + 1} - \paren {n + 1}$ | Let $N$ be a positive integer.
Then:
{{begin-eqn}}
{{eqn | l = \paren {N + 1}^{k + 1} - N^{k + 1}
| r = \sum_{i \mathop = 0}^{k + 1} \binom {k + 1} i N^i - N^{k + 1}
| c = Binomial Theorem for Integral Index
}}
{{eqn | r = \binom {k + 1} 0 + \sum_{i \mathop = 1}^k \binom {k + 1} i N^i + \binom {k + 1} {k ... | Let $n, k \in \Z_{\ge 0}$ be [[Definition:Positive Integer|positive integers]].
Let $S_k = \ds \sum_{i \mathop = 1}^n i^k$.
Then:
:$\ds \sum_{i \mathop = 1}^k \binom {k + 1} i S_i = \paren {n + 1}^{k + 1} - \paren {n + 1}$ | Let $N$ be a [[Definition:Positive Integer|positive integer]].
Then:
{{begin-eqn}}
{{eqn | l = \paren {N + 1}^{k + 1} - N^{k + 1}
| r = \sum_{i \mathop = 0}^{k + 1} \binom {k + 1} i N^i - N^{k + 1}
| c = [[Binomial Theorem for Integral Index]]
}}
{{eqn | r = \binom {k + 1} 0 + \sum_{i \mathop = 1}^k \bi... | Sum of Sequence of Binomial Coefficients by Sum of Powers of Integers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Binomial_Coefficients_by_Sum_of_Powers_of_Integers | https://proofwiki.org/wiki/Sum_of_Sequence_of_Binomial_Coefficients_by_Sum_of_Powers_of_Integers | [
"Sums of Sequences"
] | [
"Definition:Positive/Integer"
] | [
"Definition:Positive/Integer",
"Binomial Theorem/Integral Index",
"Telescoping Series/Example 1"
] |
proofwiki-14227 | Half-Range Fourier Sine Series/x by Pi minus x over 0 to Pi | Let $\map f x$ be the real function defined on $\openint 0 \pi$ as:
:$\map f x = x \paren {\pi - x}$
Then its half-range Fourier sine series can be expressed as:
:$\ds \map f x \sim \frac 8 \pi \sum_{r \mathop = 0}^\infty \frac {\sin \paren {2 r + 1} x} {\paren {2 r + 1}^3}$ | By definition of half-range Fourier sine series:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin n x$
where for all $n \in \Z_{> 0}$:
:$b_n = \ds \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$
Thus by definition of $f$:
{{begin-eqn}}
{{eqn | l = b_n
| r = \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x
... | Let $\map f x$ be the [[Definition:Real Function|real function]] defined on $\openint 0 \pi$ as:
:$\map f x = x \paren {\pi - x}$
Then its [[Definition:Half-Range Fourier Series|half-range Fourier sine series]] can be expressed as:
:$\ds \map f x \sim \frac 8 \pi \sum_{r \mathop = 0}^\infty \frac {\sin \paren {2 r ... | By definition of [[Definition:Half-Range Fourier Sine Series|half-range Fourier sine series]]:
:$\ds \map f x \sim \sum_{n \mathop = 1}^\infty b_n \sin n x$
where for all $n \in \Z_{> 0}$:
:$b_n = \ds \frac 2 \pi \int_0^\pi \map f x \sin n x \rd x$
Thus by definition of $f$:
{{begin-eqn}}
{{eqn | l = b_n
| ... | Half-Range Fourier Sine Series/x by Pi minus x over 0 to Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Sine_Series/x_by_Pi_minus_x_over_0_to_Pi | https://proofwiki.org/wiki/Half-Range_Fourier_Sine_Series/x_by_Pi_minus_x_over_0_to_Pi | [
"Examples of Half-Range Fourier Series"
] | [
"Definition:Real Function",
"Definition:Half-Range Fourier Series"
] | [
"Definition:Half-Range Fourier Sine Series",
"Linear Combination of Integrals/Definite",
"Primitive of x by Sine of a x",
"Sine of Integer Multiple of Pi",
"Primitive of x squared by Sine of a x",
"Sine of Integer Multiple of Pi",
"Cosine of Zero is One",
"Cosine of Integer Multiple of Pi",
"Definit... |
proofwiki-14228 | Sum of Reciprocals of Powers of Odd Integers Alternating in Sign | :$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s} = \frac 1 {2 \map \Gamma s} \int_0^\infty x^{s - 1} \map \sech x \rd x$
where:
:$\map \Re s > 0$
:$\Gamma$ is the gamma function
:$\sech$ is the hyperbolic secant function. | {{begin-eqn}}
{{eqn | l = \int_0^\infty x^{s - 1} \map \sech x \rd x
| r = 2 \int_0^\infty \frac { x^{s - 1} } {e^x + e^{-x} } \rd x
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = 2 \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 - \paren {- e^{-2 x} } } \rd x
}}
{{eqn | r = 2 \int_0^\infty x^{s - 1} e^{-x} \sum_{n \ma... | :$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s} = \frac 1 {2 \map \Gamma s} \int_0^\infty x^{s - 1} \map \sech x \rd x$
where:
:$\map \Re s > 0$
:$\Gamma$ is the [[Definition:Gamma Function|gamma function]]
:$\sech$ is the [[Definition:Hyperbolic Secant|hyperbolic secant function]]. | {{begin-eqn}}
{{eqn | l = \int_0^\infty x^{s - 1} \map \sech x \rd x
| r = 2 \int_0^\infty \frac { x^{s - 1} } {e^x + e^{-x} } \rd x
| c = {{Defof|Hyperbolic Secant}}
}}
{{eqn | r = 2 \int_0^\infty \frac {x^{s - 1} e^{-x} } {1 - \paren {- e^{-2 x} } } \rd x
}}
{{eqn | r = 2 \int_0^\infty x^{s - 1} e^{-x} \sum_{n \ma... | Sum of Reciprocals of Powers of Odd Integers Alternating in Sign | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Powers_of_Odd_Integers_Alternating_in_Sign | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Powers_of_Odd_Integers_Alternating_in_Sign | [
"Dirichlet Beta Function",
"Hyperbolic Secant Function",
"Definite Integrals",
"Sums of Sequences"
] | [
"Definition:Gamma Function",
"Definition:Hyperbolic Secant"
] | [
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Integration by Substitution",
"Category:Dirichlet Beta Function",
"Category:Hyperbolic Secant Function",
"Category:Definite Integrals",
"Category:Sums of Sequences"
] |
proofwiki-14229 | Equivalence of Definitions of Local Ring Homomorphism | Let $\struct {A, \mathfrak m}$ and $\struct {B, \mathfrak n}$ be commutative local rings.
Let $f : A \to B$ be a unital ring homomorphism.
{{TFAE|def = Local Ring Homomorphism}} | === 1 iff 2 ===
Follows from Image is Subset iff Subset of Preimage.
{{qed|lemma}} | Let $\struct {A, \mathfrak m}$ and $\struct {B, \mathfrak n}$ be [[Definition:Commutative Local Ring|commutative local rings]].
Let $f : A \to B$ be a [[Definition:Unital Ring Homomorphism|unital ring homomorphism]].
{{TFAE|def = Local Ring Homomorphism}} | === 1 iff 2 ===
Follows from [[Image is Subset iff Subset of Preimage]].
{{qed|lemma}} | Equivalence of Definitions of Local Ring Homomorphism | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Local_Ring_Homomorphism | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Local_Ring_Homomorphism | [
"Local Ring Homomorphisms"
] | [
"Definition:Local Ring/Commutative",
"Definition:Unital Ring Homomorphism"
] | [
"Image is Subset iff Subset of Preimage"
] |
proofwiki-14230 | Definite Integral from 0 to 1 of Power of u over 1 + Power of u | {{begin-eqn}}
{{eqn | l = \int_0^1 \dfrac {u^{a - 1} \rd u} {1 + u^d}
| r = \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {a + j d}
| c =
}}
{{eqn | r = \frac 1 a - \frac 1 {a + d} + \frac 1 {a + 2 d} - \frac 1 {a + 3 d} + \cdots
| c =
}}
{{end-eqn}}
where $a, d > 0$. | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {u^{a - 1} \rd u} {1 + u^d}
| r = \int_0^1 \frac {u^{a - 1} \rd u} {1 - \paren {-u^d} }
}}
{{eqn | r = \int_0^1 u^{a - 1} \sum_{j \mathop = 0}^\infty \paren {-1}^j u^{j d} \rd u
| c = Sum of Infinite Geometric Sequence
}}
{{eqn | r = \int_0^1 \sum_{j \mathop = 0}^\in... | {{begin-eqn}}
{{eqn | l = \int_0^1 \dfrac {u^{a - 1} \rd u} {1 + u^d}
| r = \sum_{j \mathop = 0}^\infty \frac {\paren {-1}^j} {a + j d}
| c =
}}
{{eqn | r = \frac 1 a - \frac 1 {a + d} + \frac 1 {a + 2 d} - \frac 1 {a + 3 d} + \cdots
| c =
}}
{{end-eqn}}
where $a, d > 0$. | {{begin-eqn}}
{{eqn | l = \int_0^1 \frac {u^{a - 1} \rd u} {1 + u^d}
| r = \int_0^1 \frac {u^{a - 1} \rd u} {1 - \paren {-u^d} }
}}
{{eqn | r = \int_0^1 u^{a - 1} \sum_{j \mathop = 0}^\infty \paren {-1}^j u^{j d} \rd u
| c = [[Sum of Infinite Geometric Sequence]]
}}
{{eqn | r = \int_0^1 \sum_{j \mathop = 0}... | Definite Integral from 0 to 1 of Power of u over 1 + Power of u | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Power_of_u_over_1_+_Power_of_u | https://proofwiki.org/wiki/Definite_Integral_from_0_to_1_of_Power_of_u_over_1_+_Power_of_u | [
"Examples of Definite Integrals"
] | [] | [
"Sum of Infinite Geometric Sequence",
"Fubini's Theorem",
"Primitive of Power",
"Fundamental Theorem of Calculus"
] |
proofwiki-14231 | Sum of Reciprocals of Even Powers of Odd Integers | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }
| r = \dfrac 1 {1^{2 n} } + \dfrac 1 {3^{2 n} } + \dfrac 1 {5^{2 n} } + \dfrac 1 {7^{2 n} } + \cdots
| c =
}}
{{eqn | r = \paren {-1}^{n + 1} \dfrac {B_{2 n} \paren... | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^\infty \frac 1 {j^{2 n} }
| r = \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j}^{2 n} } + \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }
| c =
}}
{{eqn | r = \frac 1 {2^{2 n} } \sum_{j \mathop = 1}^\infty \frac 1 {j^{2 n} } + \sum_{j \mathop =... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }
| r = \dfrac 1 {1^{2 n} } + \dfrac 1 {3^{2 n} } + \dfrac 1 {5^{2 n} } + \dfrac 1 {7^{2 n} } + \cdots
| c =
}}
{{eqn | r = ... | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^\infty \frac 1 {j^{2 n} }
| r = \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j}^{2 n} } + \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} }
| c =
}}
{{eqn | r = \frac 1 {2^{2 n} } \sum_{j \mathop = 1}^\infty \frac 1 {j^{2 n} } + \sum_{j \mathop =... | Sum of Reciprocals of Even Powers of Odd Integers | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Even_Powers_of_Odd_Integers | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Even_Powers_of_Odd_Integers | [
"Sum of Reciprocals of Even Powers of Odd Integers",
"Bernoulli Numbers",
"Sums of Sequences"
] | [
"Definition:Strictly Positive/Integer"
] | [
"Riemann Zeta Function at Even Integers"
] |
proofwiki-14232 | Sum of Reciprocals of Even Powers of Integers Alternating in Sign | Let $n \in \Z_{> 0}$ be a (strictly) positive integer.
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^\infty \paren {-1}^{j + 1} \frac 1 {j^{2 n} }
| r = \dfrac 1 {1^{2 n} } - \dfrac 1 {2^{2 n} } + \dfrac 1 {3^{2 n} } - \dfrac 1 {4^{2 n} } + \cdots
| c =
}}
{{eqn | r = \paren {-1}^{n + 1} \dfrac {B_{2 n} \... | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^\infty \paren {-1}^{j + 1} \frac 1 {j^{2 n} }
| r = \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} } - \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j}^{2 n} }
| c = separating odd positive terms from even negative terms
}}
{{eqn | r = \sum_{j \ma... | Let $n \in \Z_{> 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
{{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^\infty \paren {-1}^{j + 1} \frac 1 {j^{2 n} }
| r = \dfrac 1 {1^{2 n} } - \dfrac 1 {2^{2 n} } + \dfrac 1 {3^{2 n} } - \dfrac 1 {4^{2 n} } + \cdots
| c =
}}
{{eqn |... | {{begin-eqn}}
{{eqn | l = \sum_{j \mathop = 1}^\infty \paren {-1}^{j + 1} \frac 1 {j^{2 n} }
| r = \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j - 1}^{2 n} } - \sum_{j \mathop = 1}^\infty \frac 1 {\paren {2 j}^{2 n} }
| c = separating [[Definition:Odd Integer|odd]] [[Definition:Positive Real Number|posit... | Sum of Reciprocals of Even Powers of Integers Alternating in Sign | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Even_Powers_of_Integers_Alternating_in_Sign | https://proofwiki.org/wiki/Sum_of_Reciprocals_of_Even_Powers_of_Integers_Alternating_in_Sign | [
"Sum of Reciprocals of Even Powers of Integers Alternating in Sign",
"Bernoulli Numbers",
"Formulas for Pi",
"Sums of Sequences"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Bernoulli Numbers"
] | [
"Definition:Odd Integer",
"Definition:Positive/Real Number",
"Definition:Even Integer",
"Definition:Negative/Real Number",
"Sum of Reciprocals of Even Powers of Odd Integers",
"Riemann Zeta Function at Even Integers"
] |
proofwiki-14233 | Hermite's Formula for Hurwitz Zeta Function | :$\ds \map \zeta {s, q} = \frac 1 {2 q^s} + \frac { q^{1 - s} } {s - 1} + 2 \int_0^\infty \frac {\map \sin {s \arctan \frac x q} } {\paren {q^2 + x^2}^{\frac 1 2 s} \paren {e^{2 \pi x} - 1} } \rd x$
where:
:$\zeta$ is the Hurwitz zeta function
:$\map \Re s > 1$
:$\map \Re q > 0$. | {{MissingLinks|Some of these are to pages which doing exist yet}}
To prove this theorem, we can make use of Binet's Second Formula for Log Gamma:
Let $q$ be a complex number with a positive real part.
Then:
:$\ds \Ln \map \Gamma q = \paren {q - \frac 1 2} \Ln q - q + \frac 1 2 \ln 2 \pi + 2 \int_0^\infty \frac {\map \... | :$\ds \map \zeta {s, q} = \frac 1 {2 q^s} + \frac { q^{1 - s} } {s - 1} + 2 \int_0^\infty \frac {\map \sin {s \arctan \frac x q} } {\paren {q^2 + x^2}^{\frac 1 2 s} \paren {e^{2 \pi x} - 1} } \rd x$
where:
:$\zeta$ is the [[Definition:Hurwitz Zeta Function|Hurwitz zeta function]]
:$\map \Re s > 1$
:$\map \Re q > 0$. | {{MissingLinks|Some of these are to pages which doing exist yet}}
To prove this theorem, we can make use of [[Binet's Formula for Logarithm of Gamma Function/Formulation 2|Binet's Second Formula for Log Gamma]]:
Let $q$ be a [[Definition:Complex Number|complex number]] with a [[Definition:Positive Real Number|positiv... | Hermite's Formula for Hurwitz Zeta Function | https://proofwiki.org/wiki/Hermite's_Formula_for_Hurwitz_Zeta_Function | https://proofwiki.org/wiki/Hermite's_Formula_for_Hurwitz_Zeta_Function | [
"Hurwitz Zeta Function"
] | [
"Definition:Hurwitz Zeta Function"
] | [
"Binet's Formula for Logarithm of Gamma Function/Formulation 2",
"Definition:Complex Number",
"Definition:Positive/Real Number",
"Definition:Complex Number/Real Part",
"Definition:Fractional Calculus"
] |
proofwiki-14234 | Binet's Formula for Logarithm of Gamma Function/Formulation 1 | Let $z$ be a complex number with a strictly positive real part.
Then:
:$\ds \Ln \map \Gamma z = \paren {z - \frac 1 2} \Ln z - z + \frac 1 2 \ln 2 \pi + \int_0^\infty \paren {\frac 1 2 - \frac 1 t + \frac 1 {e^t - 1} } \frac {e^{-t z} } t \rd t$
where:
:$\Gamma$ is the Gamma function
:$\Ln$ is the principal branch of... | We have:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d z} } {\Ln \map \Gamma {z + 1} }
| r = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-t \paren {z + 1} } } {1 - e^{-t} } } \rd t
| c = Gauss's Integral Form of Digamma Function and {{Defof|Digamma Function}}
}}
{{eqn | r = \int_0^\infty \frac {e^{-... | Let $z$ be a [[Definition:Complex Number|complex number]] with a [[Definition:Strictly Positive|strictly positive]] [[Definition:Real Part|real part]].
Then:
:$\ds \Ln \map \Gamma z = \paren {z - \frac 1 2} \Ln z - z + \frac 1 2 \ln 2 \pi + \int_0^\infty \paren {\frac 1 2 - \frac 1 t + \frac 1 {e^t - 1} } \frac {e^{... | We have:
{{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d z} } {\Ln \map \Gamma {z + 1} }
| r = \int_0^\infty \paren {\frac {e^{-t} } t - \frac {e^{-t \paren {z + 1} } } {1 - e^{-t} } } \rd t
| c = [[Gauss's Integral Form of Digamma Function]] and {{Defof|Digamma Function}}
}}
{{eqn | r = \int_0^\infty \frac {... | Binet's Formula for Logarithm of Gamma Function/Formulation 1 | https://proofwiki.org/wiki/Binet's_Formula_for_Logarithm_of_Gamma_Function/Formulation_1 | https://proofwiki.org/wiki/Binet's_Formula_for_Logarithm_of_Gamma_Function/Formulation_1 | [
"Binet's Formula for Logarithm of Gamma Function"
] | [
"Definition:Complex Number",
"Definition:Strictly Positive",
"Definition:Complex Number/Real Part",
"Definition:Gamma Function",
"Definition:Natural Logarithm/Complex/Principal Branch",
"Definition:Natural Logarithm/Complex"
] | [
"Gauss's Integral Form of Digamma Function",
"Linear Combination of Integrals",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Exponent Combination Laws/Product of Powers",
"Linear Combination of Integrals",
"Primitive of Exponential of a x",
"Definite Integral to Infinity of Exp... |
proofwiki-14235 | Binet's Formula for Logarithm of Gamma Function/Formulation 2 | Let $z$ be a complex number with a strictly positive real part.
Then:
:$\ds \Ln \map \Gamma z = \paren {z - \frac 1 2} \Ln z - z + \frac 1 2 \ln 2 \pi + 2 \int_0^\infty \frac {\map \arctan {t / z} } {e^{2 \pi t} - 1} \rd t$
where:
:$\Gamma$ is the Gamma function
:$\Ln$ is the principal branch of the complex logarithm... | We have:
{{begin-eqn}}
{{eqn | l = \map {\dfrac {\d^2} {\d z^2} } {\Ln \map \Gamma z }
| r = \map {\dfrac {\d} {\d z} } {\map \psi z }
| c = {{Defof|Digamma Function}}
}}
{{eqn | r = \paren {-1}^{1 + 1} \map \Gamma {1 + 1} \map \zeta {1 + 1, z}
| c = Polygamma Function in terms of Hurwitz Zeta Functio... | Let $z$ be a [[Definition:Complex Number|complex number]] with a [[Definition:Strictly Positive|strictly positive]] [[Definition:Real Part|real part]].
Then:
:$\ds \Ln \map \Gamma z = \paren {z - \frac 1 2} \Ln z - z + \frac 1 2 \ln 2 \pi + 2 \int_0^\infty \frac {\map \arctan {t / z} } {e^{2 \pi t} - 1} \rd t$
wher... | We have:
{{begin-eqn}}
{{eqn | l = \map {\dfrac {\d^2} {\d z^2} } {\Ln \map \Gamma z }
| r = \map {\dfrac {\d} {\d z} } {\map \psi z }
| c = {{Defof|Digamma Function}}
}}
{{eqn | r = \paren {-1}^{1 + 1} \map \Gamma {1 + 1} \map \zeta {1 + 1, z}
| c = [[Polygamma Function in terms of Hurwitz Zeta Funct... | Binet's Formula for Logarithm of Gamma Function/Formulation 2 | https://proofwiki.org/wiki/Binet's_Formula_for_Logarithm_of_Gamma_Function/Formulation_2 | https://proofwiki.org/wiki/Binet's_Formula_for_Logarithm_of_Gamma_Function/Formulation_2 | [
"Binet's Formula for Logarithm of Gamma Function"
] | [
"Definition:Complex Number",
"Definition:Strictly Positive",
"Definition:Complex Number/Real Part",
"Definition:Gamma Function",
"Definition:Natural Logarithm/Complex/Principal Branch",
"Definition:Natural Logarithm/Complex"
] | [
"Polygamma Function in terms of Hurwitz Zeta Function",
"Abel-Plana Formula",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Definition:Holomorphic Function",
"Definition:Hurwitz Zeta Function",
"Definition:Convergent Series",
"Definition:Uniform Convergence",
"Definition:Subdivision of Int... |
proofwiki-14236 | Lagrange's Trigonometric Identities/Sine/Sine Form | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \sin k x
| r = \sin 0 + \sin x + \sin 2 x + \sin 3 x + \cdots + \sin n x
| c =
}}
{{eqn | r = \frac {\sin \frac {\paren {n + 1} x} 2 \sin \frac {n x} 2} {\sin \frac x 2}
| c =
}}
{{end-eqn}}
where $x$ is not an integer multiple of $2 \pi$. | By Werner Formula for Sine by Sine:
:$2 \sin \alpha \sin \beta = \map \cos {\alpha - \beta} - \map \cos {\alpha + \beta}$
Thus we establish the following sequence of identities:
{{begin-eqn}}
{{eqn | l = 2 \sin 0 \sin \frac x 2
| r = 0
| c =
}}
{{eqn | l = 2 \sin x \sin \frac x 2
| r = \cos \frac x 2... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \sin k x
| r = \sin 0 + \sin x + \sin 2 x + \sin 3 x + \cdots + \sin n x
| c =
}}
{{eqn | r = \frac {\sin \frac {\paren {n + 1} x} 2 \sin \frac {n x} 2} {\sin \frac x 2}
| c =
}}
{{end-eqn}}
where $x$ is not an [[Definition:Integer Multiple|integer m... | By [[Werner Formula for Sine by Sine]]:
:$2 \sin \alpha \sin \beta = \map \cos {\alpha - \beta} - \map \cos {\alpha + \beta}$
Thus we establish the following sequence of identities:
{{begin-eqn}}
{{eqn | l = 2 \sin 0 \sin \frac x 2
| r = 0
| c =
}}
{{eqn | l = 2 \sin x \sin \frac x 2
| r = \cos \fr... | Lagrange's Trigonometric Identities/Sine/Sine Form/Proof 1 | https://proofwiki.org/wiki/Lagrange's_Trigonometric_Identities/Sine/Sine_Form | https://proofwiki.org/wiki/Lagrange's_Trigonometric_Identities/Sine/Sine_Form/Proof_1 | [
"Sine Form of Lagrange's Sine Identity",
"Lagrange's Sine Identity",
"Lagrange's Trigonometric Identities",
"Sine Function",
"Telescoping Series"
] | [
"Definition:Integral Multiple/Real Numbers"
] | [
"Werner Formulas/Sine by Sine",
"Definition:Telescoping Series",
"Prosthaphaeresis Formulas/Cosine minus Cosine",
"Sine Function is Odd"
] |
proofwiki-14237 | Lagrange's Trigonometric Identities/Sine/Sine Form | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \sin k x
| r = \sin 0 + \sin x + \sin 2 x + \sin 3 x + \cdots + \sin n x
| c =
}}
{{eqn | r = \frac {\sin \frac {\paren {n + 1} x} 2 \sin \frac {n x} 2} {\sin \frac x 2}
| c =
}}
{{end-eqn}}
where $x$ is not an integer multiple of $2 \pi$. | Let $x$ be a real number that is not a integer multiple of $2 \pi$.
Let $k$ be a non-negative integer.
We have, from Euler's Formula:
:$\map \exp {i k x} = i \sin k x + \cos k x$
Summing from $k = 0$ to $k = n$, we have:
:$\ds \sum_{k \mathop = 0}^n \map \exp {i k x} = i \sum_{k \mathop = 0}^n \sin k x + \sum_{k \math... | {{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n \sin k x
| r = \sin 0 + \sin x + \sin 2 x + \sin 3 x + \cdots + \sin n x
| c =
}}
{{eqn | r = \frac {\sin \frac {\paren {n + 1} x} 2 \sin \frac {n x} 2} {\sin \frac x 2}
| c =
}}
{{end-eqn}}
where $x$ is not an [[Definition:Integer Multiple|integer m... | Let $x$ be a [[Definition:Real Number|real number]] that is not a [[Definition:Integer Multiple|integer multiple]] of $2 \pi$.
Let $k$ be a [[Definition:Non-Negative Integer|non-negative integer]].
We have, from [[Euler's Formula]]:
:$\map \exp {i k x} = i \sin k x + \cos k x$
Summing from $k = 0$ to $k = n$, we ha... | Lagrange's Trigonometric Identities/Sine/Sine Form/Proof 2 | https://proofwiki.org/wiki/Lagrange's_Trigonometric_Identities/Sine/Sine_Form | https://proofwiki.org/wiki/Lagrange's_Trigonometric_Identities/Sine/Sine_Form/Proof_2 | [
"Sine Form of Lagrange's Sine Identity",
"Lagrange's Sine Identity",
"Lagrange's Trigonometric Identities",
"Sine Function",
"Telescoping Series"
] | [
"Definition:Integral Multiple/Real Numbers"
] | [
"Definition:Real Number",
"Definition:Integral Multiple/Real Numbers",
"Definition:Positive/Integer",
"Euler's Formula",
"Definition:Real Number",
"Sum of Exponential of i k x"
] |
proofwiki-14238 | Sum of Infinite Series of Product of Power and Cosine | Let $r \in \R$ such that $\size r < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty r^k \cos k x
| r = 1 + r \cos x + r^2 \cos 2 x + r^3 \cos 3 x + \cdots
| c =
}}
{{eqn| r = \dfrac {1 - r \cos x} {1 - 2 r \cos x + r^2}
| c =
}}
{{end-eqn}} | From Euler's Formula:
:$e^{i \theta} = \cos \theta + i \sin \theta$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty r^k \cos k x
| r = \map \Re {\sum_{k \mathop = 0}^\infty r^k e^{i k x} }
| c =
}}
{{eqn | r = \map \Re {\sum_{k \mathop = 0}^\infty \paren {r e^{i x} }^k}
| c =
}}
{{eqn |... | Let $r \in \R$ such that $\size r < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty r^k \cos k x
| r = 1 + r \cos x + r^2 \cos 2 x + r^3 \cos 3 x + \cdots
| c =
}}
{{eqn| r = \dfrac {1 - r \cos x} {1 - 2 r \cos x + r^2}
| c =
}}
{{end-eqn}} | From [[Euler's Formula]]:
:$e^{i \theta} = \cos \theta + i \sin \theta$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^\infty r^k \cos k x
| r = \map \Re {\sum_{k \mathop = 0}^\infty r^k e^{i k x} }
| c =
}}
{{eqn | r = \map \Re {\sum_{k \mathop = 0}^\infty \paren {r e^{i x} }^k}
| c =
}}
{... | Sum of Infinite Series of Product of Power and Cosine | https://proofwiki.org/wiki/Sum_of_Infinite_Series_of_Product_of_Power_and_Cosine | https://proofwiki.org/wiki/Sum_of_Infinite_Series_of_Product_of_Power_and_Cosine | [
"Cosine Function",
"Trigonometric Series"
] | [] | [
"Euler's Formula",
"Sum of Infinite Geometric Sequence"
] |
proofwiki-14239 | Sum of Series of Product of Power and Cosine | Let $r \in \R$ such that $\size r < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n r^k \map \cos {k x}
| r = 1 + r \cos x + r^2 \cos 2 x + r^3 \cos 3 x + \cdots + r^n \cos n x
| c =
}}
{{eqn | r = \dfrac {r^{n + 2} \cos n x - r^{n + 1} \map \cos {n + 1} x - r \cos x + 1} {1 - 2 r \cos x + r^2}
... | From Euler's Formula:
:$e^{i \theta} = \cos \theta + i \sin \theta$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n r^k \map \cos {k x}
| r = \map \Re {\sum_{k \mathop = 0}^n r^k e^{i k x} }
| c =
}}
{{eqn | r = \map \Re {\sum_{k \mathop = 0}^n \paren {r e^{i x} }^k}
| c =
}}
{{eqn | r = \ma... | Let $r \in \R$ such that $\size r < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n r^k \map \cos {k x}
| r = 1 + r \cos x + r^2 \cos 2 x + r^3 \cos 3 x + \cdots + r^n \cos n x
| c =
}}
{{eqn | r = \dfrac {r^{n + 2} \cos n x - r^{n + 1} \map \cos {n + 1} x - r \cos x + 1} {1 - 2 r \cos x + r^2}... | From [[Euler's Formula]]:
:$e^{i \theta} = \cos \theta + i \sin \theta$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 0}^n r^k \map \cos {k x}
| r = \map \Re {\sum_{k \mathop = 0}^n r^k e^{i k x} }
| c =
}}
{{eqn | r = \map \Re {\sum_{k \mathop = 0}^n \paren {r e^{i x} }^k}
| c =
}}
{{eqn | r... | Sum of Series of Product of Power and Cosine | https://proofwiki.org/wiki/Sum_of_Series_of_Product_of_Power_and_Cosine | https://proofwiki.org/wiki/Sum_of_Series_of_Product_of_Power_and_Cosine | [
"Cosine Function",
"Trigonometric Series"
] | [] | [
"Euler's Formula",
"Sum of Infinite Geometric Sequence",
"Euler's Formula"
] |
proofwiki-14240 | Sum of Series of Product of Power and Sine | Let $r \in \R$ such that $\size r < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n r^k \map \sin {k x}
| r = r \sin x + r^2 \sin 2 x + r^3 \sin 3 x + \cdots + r^n \sin n x
| c =
}}
{{eqn | r = \dfrac {r \sin x - r^{n + 1} \map \sin {n + 1} x + r^{n + 2} \sin n x} {1 - 2 r \cos x + r^2}
| ... | From Euler's Formula:
:$e^{i \theta} = \cos \theta + i \sin \theta$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n r^k \map \sin {k x}
| r = \map \Im {\sum_{k \mathop = 1}^n r^k e^{i k x} }
| c =
}}
{{eqn | r = \map \Im {\sum_{k \mathop = 0}^n \paren {r e^{i x} }^n}
| c = as $\map \Im {e^{i ... | Let $r \in \R$ such that $\size r < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n r^k \map \sin {k x}
| r = r \sin x + r^2 \sin 2 x + r^3 \sin 3 x + \cdots + r^n \sin n x
| c =
}}
{{eqn | r = \dfrac {r \sin x - r^{n + 1} \map \sin {n + 1} x + r^{n + 2} \sin n x} {1 - 2 r \cos x + r^2}
|... | From [[Euler's Formula]]:
:$e^{i \theta} = \cos \theta + i \sin \theta$
Hence:
{{begin-eqn}}
{{eqn | l = \sum_{k \mathop = 1}^n r^k \map \sin {k x}
| r = \map \Im {\sum_{k \mathop = 1}^n r^k e^{i k x} }
| c =
}}
{{eqn | r = \map \Im {\sum_{k \mathop = 0}^n \paren {r e^{i x} }^n}
| c = as $\map \Im ... | Sum of Series of Product of Power and Sine | https://proofwiki.org/wiki/Sum_of_Series_of_Product_of_Power_and_Sine | https://proofwiki.org/wiki/Sum_of_Series_of_Product_of_Power_and_Sine | [
"Sine Function",
"Trigonometric Series"
] | [] | [
"Euler's Formula",
"Sum of Infinite Geometric Sequence",
"Euler's Formula"
] |
proofwiki-14241 | Power Series Expansion for Reciprocal of 1 + x | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {1 + x}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k x^k
| c =
}}
{{eqn | r = 1 - x + x^2 - x^3 + x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {1 + x}
| r = \frac 1 {1 - \paren {-x} }
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \paren {-x}^k
| c = Sum of Infinite Geometric Sequence
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \paren {-1}^k x^k
| c =
}}
{{end-eqn}}
{{qed}} | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {1 + x}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k x^k
| c =
}}
{{eqn | r = 1 - x + x^2 - x^3 + x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {1 + x}
| r = \frac 1 {1 - \paren {-x} }
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \paren {-x}^k
| c = [[Sum of Infinite Geometric Sequence]]
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \paren {-1}^k x^k
| c =
}}
{{end-eqn}}
{{qed}} | Power Series Expansion for Reciprocal of 1 + x/Proof 1 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_1_+_x/Proof_1 | [
"Power Series Expansion for Reciprocal of 1 + x",
"Examples of Power Series"
] | [] | [
"Sum of Infinite Geometric Sequence"
] |
proofwiki-14242 | Power Series Expansion for Reciprocal of 1 + x | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {1 + x}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k x^k
| c =
}}
{{eqn | r = 1 - x + x^2 - x^3 + x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {1 + x}
| r = \paren {1 + x}^{-1}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^{\underline k} } {k!} x^k
| c = General Binomial Theorem
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {\paren {-1} - j}... | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {1 + x}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k x^k
| c =
}}
{{eqn | r = 1 - x + x^2 - x^3 + x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {1 + x}
| r = \paren {1 + x}^{-1}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^{\underline k} } {k!} x^k
| c = [[General Binomial Theorem]]
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {\paren {-1} ... | Power Series Expansion for Reciprocal of 1 + x/Proof 2 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_1_+_x/Proof_2 | [
"Power Series Expansion for Reciprocal of 1 + x",
"Examples of Power Series"
] | [] | [
"Binomial Theorem/General Binomial Theorem",
"Translation of Index Variable of Product"
] |
proofwiki-14243 | Power Series Expansion for Reciprocal of Square of 1 + x | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\paren {1 + x}^2}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {k + 1} x^k
| c =
}}
{{eqn | r = 1 - 2 x + 3 x^2 - 4 x^3 + 5 x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {1 + x}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k x^k
| c = Power Series Expansion for $\dfrac 1 {1 + x}$
}}
{{eqn | ll= \leadsto
| l = \frac \d {\d x} \frac 1 {1 + x}
| r = \frac \d {\d x} \sum_{k \mathop = 0}^\infty \paren {-1}^k x^k
| c =
}}
{{eq... | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\paren {1 + x}^2}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {k + 1} x^k
| c =
}}
{{eqn | r = 1 - 2 x + 3 x^2 - 4 x^3 + 5 x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {1 + x}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k x^k
| c = [[Power Series Expansion for Reciprocal of 1 + x|Power Series Expansion for $\dfrac 1 {1 + x}$]]
}}
{{eqn | ll= \leadsto
| l = \frac \d {\d x} \frac 1 {1 + x}
| r = \frac \d {\d x} \sum_{k \mathop... | Power Series Expansion for Reciprocal of Square of 1 + x/Proof 1 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Square_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Square_of_1_+_x/Proof_1 | [
"Examples of Power Series",
"Power Series Expansion for Reciprocal of Square of 1 + x"
] | [] | [
"Power Series Expansion for Reciprocal of 1 + x",
"Definition:Differentiation",
"Translation of Index Variable of Product"
] |
proofwiki-14244 | Power Series Expansion for Reciprocal of Square of 1 + x | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\paren {1 + x}^2}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {k + 1} x^k
| c =
}}
{{eqn | r = 1 - 2 x + 3 x^2 - 4 x^3 + 5 x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {\paren {1 + x} }
| r = \paren {1 + x}^{-2}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-2}^{\underline k} } {k!} x^k
| c = General Binomial Theorem
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {\paren... | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\paren {1 + x}^2}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {k + 1} x^k
| c =
}}
{{eqn | r = 1 - 2 x + 3 x^2 - 4 x^3 + 5 x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {\paren {1 + x} }
| r = \paren {1 + x}^{-2}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-2}^{\underline k} } {k!} x^k
| c = [[General Binomial Theorem]]
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {\p... | Power Series Expansion for Reciprocal of Square of 1 + x/Proof 2 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Square_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Square_of_1_+_x/Proof_2 | [
"Examples of Power Series",
"Power Series Expansion for Reciprocal of Square of 1 + x"
] | [] | [
"Binomial Theorem/General Binomial Theorem",
"Translation of Index Variable of Product"
] |
proofwiki-14245 | Frullani's Integral | :$\ds \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x = \paren {\map f \infty - \map f 0} \ln \frac a b$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x
| r = \int_0^\infty \intlimits {\frac {\map f {x t} } x} {t = b} a \rd x
}}
{{eqn | r = \int_0^\infty \int_b^a \map {f'} {x t} \rd t \rd x
| c = Fundamental Theorem of Calculus
}}
{{eqn | r = \int_b^a \int_0^\infty \map {f'} {x t} \... | :$\ds \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x = \paren {\map f \infty - \map f 0} \ln \frac a b$ | {{begin-eqn}}
{{eqn | l = \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x
| r = \int_0^\infty \intlimits {\frac {\map f {x t} } x} {t = b} a \rd x
}}
{{eqn | r = \int_0^\infty \int_b^a \map {f'} {x t} \rd t \rd x
| c = [[Fundamental Theorem of Calculus]]
}}
{{eqn | r = \int_b^a \int_0^\infty \map {f'} {x ... | Frullani's Integral | https://proofwiki.org/wiki/Frullani's_Integral | https://proofwiki.org/wiki/Frullani's_Integral | [
"Definite Integrals"
] | [] | [
"Fundamental Theorem of Calculus",
"Fubini's Theorem",
"Fundamental Theorem of Calculus",
"Primitive of Reciprocal",
"Fundamental Theorem of Calculus",
"Difference of Logarithms"
] |
proofwiki-14246 | Power Series Expansion for Reciprocal of Cube of 1 + x | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\paren {1 + x}^3}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {k + 2} \paren {k + 1} } 2 x^k
| c =
}}
{{eqn | r = 1 - 3 x + 6 x^2 - 10 x^3 + 15 x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {\paren {1 + x}^2}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {k + 1} x^k
| c = Power Series Expansion of $\dfrac 1 {\paren {1 + x}^2}$
}}
{{eqn | ll= \leadsto
| l = \frac \d {\d x} \frac 1 {\paren {1 + x}^2}
| r = \frac \d {\d x} \sum_{k \mathop = 0... | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\paren {1 + x}^3}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {k + 2} \paren {k + 1} } 2 x^k
| c =
}}
{{eqn | r = 1 - 3 x + 6 x^2 - 10 x^3 + 15 x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {\paren {1 + x}^2}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \paren {k + 1} x^k
| c = [[Power Series Expansion for Reciprocal of Square of 1 + x|Power Series Expansion of $\dfrac 1 {\paren {1 + x}^2}$]]
}}
{{eqn | ll= \leadsto
| l = \frac \d {\d x} \frac 1 {\par... | Power Series Expansion for Reciprocal of Cube of 1 + x/Proof 1 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Cube_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Cube_of_1_+_x/Proof_1 | [
"Examples of Power Series",
"Power Series Expansion for Reciprocal of Cube of 1 + x"
] | [] | [
"Power Series Expansion for Reciprocal of Square of 1 + x",
"Definition:Differentiation",
"Translation of Index Variable of Product"
] |
proofwiki-14247 | Power Series Expansion for Reciprocal of Cube of 1 + x | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\paren {1 + x}^3}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {k + 2} \paren {k + 1} } 2 x^k
| c =
}}
{{eqn | r = 1 - 3 x + 6 x^2 - 10 x^3 + 15 x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {\paren {1 + x}^3 }
| r = \paren {1 + x}^{-3}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-3}^{\underline k} } {k!} x^k
| c = General Binomial Theorem
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {\par... | Let $x \in \R$ such that $-1 < x < 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\paren {1 + x}^3}
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {k + 2} \paren {k + 1} } 2 x^k
| c =
}}
{{eqn | r = 1 - 3 x + 6 x^2 - 10 x^3 + 15 x^4 - \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \frac 1 {\paren {1 + x}^3 }
| r = \paren {1 + x}^{-3}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-3}^{\underline k} } {k!} x^k
| c = [[General Binomial Theorem]]
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {... | Power Series Expansion for Reciprocal of Cube of 1 + x/Proof 2 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Cube_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Cube_of_1_+_x/Proof_2 | [
"Examples of Power Series",
"Power Series Expansion for Reciprocal of Cube of 1 + x"
] | [] | [
"Binomial Theorem/General Binomial Theorem",
"Translation of Index Variable of Product"
] |
proofwiki-14248 | Power Series Expansion for Reciprocal of Square Root of 1 + x | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\sqrt {1 + x} }
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {2 k}!} {\paren {2^k k!}^2} x^k
| c =
}}
{{eqn | r = 1 - \frac 1 2 x + \frac {1 \times 3} {2 \times 4} x^2 - \frac {1 \times 3 \times 5} {2 \times... | {{begin-eqn}}
{{eqn | l = \frac 1 {\sqrt {1 + x} }
| r = \paren {1 + x}^{-\frac 1 2}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-\frac 1 2}^{\underline k} } {k!} x^k
| c = General Binomial Theorem
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1}... | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
{{begin-eqn}}
{{eqn | l = \dfrac 1 {\sqrt {1 + x} }
| r = \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {2 k}!} {\paren {2^k k!}^2} x^k
| c =
}}
{{eqn | r = 1 - \frac 1 2 x + \frac {1 \times 3} {2 \times 4} x^2 - \frac {1 \times 3 \times 5} {2 \tim... | {{begin-eqn}}
{{eqn | l = \frac 1 {\sqrt {1 + x} }
| r = \paren {1 + x}^{-\frac 1 2}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-\frac 1 2}^{\underline k} } {k!} x^k
| c = [[General Binomial Theorem]]
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k ... | Power Series Expansion for Reciprocal of Square Root of 1 + x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Square_Root_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Square_Root_of_1_+_x | [
"Examples of Power Series"
] | [] | [
"Binomial Theorem/General Binomial Theorem",
"Translation of Index Variable of Product",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-14249 | Power Series Expansion for Square Root of 1 + x | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sqrt {1 + x}
| r = 1 + \sum_{k \mathop = 1}^\infty \paren {-1}^{k - 1} \frac {\paren {2 \paren {k - 1} }!} {2^{2 k - 1} k! \paren {k - 1}!} x^k
| c =
}}
{{eqn | r = 1 + \frac 1 2 x - \frac 1 {2 \times 4} x^2 + \frac {1 \times 3} {2 \... | {{begin-eqn}}
{{eqn | l = \sqrt {1 + x}
| r = \paren {1 + x}^{\frac 1 2}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {\frac 1 2}^{\underline k} } {k!} x^k
| c = General Binomial Theorem
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {\fra... | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
{{begin-eqn}}
{{eqn | l = \sqrt {1 + x}
| r = 1 + \sum_{k \mathop = 1}^\infty \paren {-1}^{k - 1} \frac {\paren {2 \paren {k - 1} }!} {2^{2 k - 1} k! \paren {k - 1}!} x^k
| c =
}}
{{eqn | r = 1 + \frac 1 2 x - \frac 1 {2 \times 4} x^2 + \frac {1 \times 3} {2... | {{begin-eqn}}
{{eqn | l = \sqrt {1 + x}
| r = \paren {1 + x}^{\frac 1 2}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {\frac 1 2}^{\underline k} } {k!} x^k
| c = [[General Binomial Theorem]]
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {... | Power Series Expansion for Square Root of 1 + x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Square_Root_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Square_Root_of_1_+_x | [
"Examples of Power Series"
] | [] | [
"Binomial Theorem/General Binomial Theorem",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator"
] |
proofwiki-14250 | Power Series Expansion for Reciprocal of Cube Root of 1 + x | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
:$\dfrac 1 {\sqrt [3] {1 + x} } = 1 - \dfrac 1 3 x + \dfrac {1 \times 4} {3 \times 6} x^2 - \dfrac {1 \times 4 \times 7} {3 \times 6 \times 9} x^3 + \cdots$ | {{begin-eqn}}
{{eqn | l = \frac 1 {\sqrt [3] {1 + x} }
| r = \paren {1 + x}^{-\frac 1 3}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-\frac 1 3}^{\underline k} } {k!} x^k
| c = General Binomial Theorem
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k ... | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
:$\dfrac 1 {\sqrt [3] {1 + x} } = 1 - \dfrac 1 3 x + \dfrac {1 \times 4} {3 \times 6} x^2 - \dfrac {1 \times 4 \times 7} {3 \times 6 \times 9} x^3 + \cdots$ | {{begin-eqn}}
{{eqn | l = \frac 1 {\sqrt [3] {1 + x} }
| r = \paren {1 + x}^{-\frac 1 3}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {-\frac 1 3}^{\underline k} } {k!} x^k
| c = [[General Binomial Theorem]]
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}... | Power Series Expansion for Reciprocal of Cube Root of 1 + x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Cube_Root_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Reciprocal_of_Cube_Root_of_1_+_x | [
"Examples of Power Series"
] | [] | [
"Binomial Theorem/General Binomial Theorem",
"Translation of Index Variable of Product"
] |
proofwiki-14251 | Power Series Expansion for Cube Root of 1 + x | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
:$\sqrt [3] {1 + x} = 1 + \dfrac 1 3 x - \dfrac 2 {3 \times 6} x^2 + \dfrac {2 \times 5} {3 \times 6 \times 9} x^3 - \cdots$ | {{begin-eqn}}
{{eqn | l = \sqrt [3] {1 + x}
| r = \paren {1 + x}^{\frac 1 3}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {\frac 1 3}^{\underline k} } {k!} x^k
| c = General Binomial Theorem
}}
{{eqn | r = 1 + \sum_{k \mathop = 1}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \par... | Let $x \in \R$ such that $-1 < x \le 1$.
Then:
:$\sqrt [3] {1 + x} = 1 + \dfrac 1 3 x - \dfrac 2 {3 \times 6} x^2 + \dfrac {2 \times 5} {3 \times 6 \times 9} x^3 - \cdots$ | {{begin-eqn}}
{{eqn | l = \sqrt [3] {1 + x}
| r = \paren {1 + x}^{\frac 1 3}
| c =
}}
{{eqn | r = \sum_{k \mathop = 0}^\infty \frac {\paren {\frac 1 3}^{\underline k} } {k!} x^k
| c = [[General Binomial Theorem]]
}}
{{eqn | r = 1 + \sum_{k \mathop = 1}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} ... | Power Series Expansion for Cube Root of 1 + x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Cube_Root_of_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Cube_Root_of_1_+_x | [
"Examples of Power Series"
] | [] | [
"Binomial Theorem/General Binomial Theorem"
] |
proofwiki-14252 | Power Series Expansion for General Exponential Function | Let $a \in \R_{> 0}$ be a (strictly) positive real number.
Then:
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = a^x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {x \ln a}^n} {n!}
| c =
}}
{{eqn | r = 1 + x \ln a + \frac {\paren {x \ln a}^2} {2!} + \frac {\paren {x \ln a}^3} {3!} + \cdots
... | By definition of a power to a real number:
:$a^x = \map \exp {x \ln a}$
As $x \ln a$ is itself a real number, we can use Power Series Expansion for Exponential Function:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \exp x
| r = \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}
| c =
}}
{{eqn | r = 1 +... | Let $a \in \R_{> 0}$ be a [[Definition:Strictly Positive Real Number|(strictly) positive real number]].
Then:
Then:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = a^x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {x \ln a}^n} {n!}
| c =
}}
{{eqn | r = 1 + x \ln a + \frac {\paren {x \ln a}^2} {... | By definition of a [[Definition:Power (Algebra)/Real Number/Definition 1|power to a real number]]:
:$a^x = \map \exp {x \ln a}$
As $x \ln a$ is itself a [[Definition:Real Number|real number]], we can use [[Power Series Expansion for Exponential Function]]:
{{begin-eqn}}
{{eqn | q = \forall x \in \R
| l = \exp x... | Power Series Expansion for General Exponential Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_General_Exponential_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_General_Exponential_Function | [
"Exponential Function",
"Examples of Power Series"
] | [
"Definition:Strictly Positive/Real Number"
] | [
"Definition:Power (Algebra)/Real Number/Definition 1",
"Definition:Real Number",
"Power Series Expansion for Exponential Function"
] |
proofwiki-14253 | Power Series Expansion for Half Logarithm of 1 + x over 1 - x | {{begin-eqn}}
{{eqn | l = \frac 1 2 \map \ln {\frac {1 + x} {1 - x} }
| r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots
}}
{{end-eqn}}
valid for all $x \in \R$ such that $-1 < x < 1$. | From Power Series Expansion for $\map \ln {1 + x}$:
:$(1): \quad \ds \map \ln {1 + x} = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n$
for $-1 < x \le 1$.
From Power Series Expansion for $\map \ln {1 - x}$:
:$(2): \quad \ds \map \ln {1 - x} = - \sum_{n \mathop = 1}^\infty \frac {x^n} n$
for $-1 < x < 1$... | {{begin-eqn}}
{{eqn | l = \frac 1 2 \map \ln {\frac {1 + x} {1 - x} }
| r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots
}}
{{end-eqn}}
valid for all $x \in \R$ such that $-1 < x < 1$. | From [[Power Series Expansion for Logarithm of 1 + x|Power Series Expansion for $\map \ln {1 + x}$]]:
:$(1): \quad \ds \map \ln {1 + x} = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n$
for $-1 < x \le 1$.
From [[Power Series Expansion for Logarithm of 1 - x|Power Series Expansion for $\map \ln {1 - x}... | Power Series Expansion for Half Logarithm of 1 + x over 1 - x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Half_Logarithm_of_1_+_x_over_1_-_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Half_Logarithm_of_1_+_x_over_1_-_x | [
"Power Series Expansion for Half Logarithm of 1 + x over 1 - x",
"Examples of Power Series",
"Logarithms"
] | [] | [
"Power Series Expansion for Logarithm of 1 + x",
"Power Series Expansion for Logarithm of 1 - x",
"Definition:Odd Integer",
"Definition:Even Integer",
"Definition:Even Integer",
"Difference of Logarithms"
] |
proofwiki-14254 | Power Series Expansion for Logarithm of x/Formulation 1 | {{begin-eqn}}
{{eqn | l = \ln x
| r = 2 \paren {\sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} \paren {\frac {x - 1} {x + 1} }^{2 n + 1} }
}}
{{eqn | r = 2 \paren {\frac {x - 1} {x + 1} + \frac 1 3 \paren {\frac {x - 1} {x + 1} }^3 + \frac 1 5 \paren {\frac {x - 1} {x + 1} }^5 + \cdots}
}}
{{end-eqn}}
valid for a... | From Power Series Expansion for $\dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$:
:$(1): \quad \ds \frac 1 2 \map \ln {\frac {1 + x} {1 - x} } = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}$
for $-1 < x < 1$.
Let $z = \dfrac {1 + x} {1 - x}$.
Then:
{{begin-eqn}}
{{eqn | l = z
| r = \dfrac {1 + x} {1 ... | {{begin-eqn}}
{{eqn | l = \ln x
| r = 2 \paren {\sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} \paren {\frac {x - 1} {x + 1} }^{2 n + 1} }
}}
{{eqn | r = 2 \paren {\frac {x - 1} {x + 1} + \frac 1 3 \paren {\frac {x - 1} {x + 1} }^3 + \frac 1 5 \paren {\frac {x - 1} {x + 1} }^5 + \cdots}
}}
{{end-eqn}}
valid for ... | From [[Power Series Expansion for Half Logarithm of 1 + x over 1 - x|Power Series Expansion for $\dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$]]:
:$(1): \quad \ds \frac 1 2 \map \ln {\frac {1 + x} {1 - x} } = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}$
for $-1 < x < 1$.
Let $z = \dfrac {1 + x} {1 - x... | Power Series Expansion for Logarithm of x/Formulation 1 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_x/Formulation_1 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_x/Formulation_1 | [
"Power Series Expansion for Logarithm of x"
] | [] | [
"Power Series Expansion for Half Logarithm of 1 + x over 1 - x"
] |
proofwiki-14255 | Power Series Expansion for Logarithm of x/Formulation 2 | {{begin-eqn}}
{{eqn | l = \ln x
| r = \sum_{n \mathop = 1}^\infty \dfrac 1 n \paren {\frac {x - 1} x}^n
}}
{{eqn | r = \frac {x - 1} x + \frac 1 2 \paren {\frac {x - 1} x}^2 + \frac 1 3 \paren {\frac {x - 1} x}^3 + \cdots
}}
{{end-eqn}}
valid for all $x \in \R$ such that $x \ge \dfrac 1 2$. | From Power Series Expansion for $\map \ln {1 - x}$:
{{begin-eqn}}
{{eqn | l = \map \ln {1 - x}
| r = -\sum_{n \mathop = 1}^\infty \frac {x^n} n
| c = for $-1 \le x < 1$.
}}
{{eqn | n = 1
| ll= \leadsto
| l = \map \ln {\frac 1 {1 - x} }
| r = \sum_{n \mathop = 1}^\infty \frac {x^n} n
... | {{begin-eqn}}
{{eqn | l = \ln x
| r = \sum_{n \mathop = 1}^\infty \dfrac 1 n \paren {\frac {x - 1} x}^n
}}
{{eqn | r = \frac {x - 1} x + \frac 1 2 \paren {\frac {x - 1} x}^2 + \frac 1 3 \paren {\frac {x - 1} x}^3 + \cdots
}}
{{end-eqn}}
valid for all $x \in \R$ such that $x \ge \dfrac 1 2$. | From [[Power Series Expansion for Logarithm of 1 - x|Power Series Expansion for $\map \ln {1 - x}$]]:
{{begin-eqn}}
{{eqn | l = \map \ln {1 - x}
| r = -\sum_{n \mathop = 1}^\infty \frac {x^n} n
| c = for $-1 \le x < 1$.
}}
{{eqn | n = 1
| ll= \leadsto
| l = \map \ln {\frac 1 {1 - x} }
| r... | Power Series Expansion for Logarithm of x/Formulation 2 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_x/Formulation_2 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_x/Formulation_2 | [
"Power Series Expansion for Logarithm of x"
] | [] | [
"Power Series Expansion for Logarithm of 1 - x",
"Logarithm of Reciprocal"
] |
proofwiki-14256 | Power Series Expansion for Cotangent Function | The (real) cotangent function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \cot x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 x - \frac x 3 - \frac {x^3} {45} - \frac {2 x^5} {945} + \frac {x^7} {4725} - \c... | {{begin-eqn}}
{{eqn | l = \cot x
| r = i \frac {e^{i x} + e^{- i x} } {e^{i x} - e^{- i x} }
| c = Euler's Cotangent Identity
}}
{{eqn | r = i \frac {e^{2 i x} + 1 } {e^{2 i x} - 1 }
| c = multiplying top and bottom by $e^{ix}$
}}
{{eqn | r = i \paren {\frac {e^{2 i x} - 1 + 2 } {e^{2 i x} - 1} }
... | The [[Definition:Real Cotangent Function|(real) cotangent function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \cot x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 x - \frac x... | {{begin-eqn}}
{{eqn | l = \cot x
| r = i \frac {e^{i x} + e^{- i x} } {e^{i x} - e^{- i x} }
| c = [[Euler's Cotangent Identity]]
}}
{{eqn | r = i \frac {e^{2 i x} + 1 } {e^{2 i x} - 1 }
| c = multiplying [[Definition:Numerator|top]] and [[Definition:Denominator|bottom]] by $e^{ix}$
}}
{{eqn | r = i \... | Power Series Expansion for Cotangent Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Cotangent_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Cotangent_Function | [
"Power Series Expansion for Cotangent Function",
"Examples of Power Series",
"Bernoulli Numbers",
"Cotangent Function"
] | [
"Definition:Cotangent/Real Function",
"Definition:Taylor Series",
"Definition:Bernoulli Numbers",
"Definition:Convergent Series"
] | [
"Euler's Cotangent Identity",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Odd Bernoulli Numbers Vanish",
"Definition:Complex Number/Definition 1",
"Combination Theorem for Limits of Functions/Real",
"Asympto... |
proofwiki-14257 | Power Series Expansion for Secant Function | The (real) secant function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \sec x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {E_{2 n} x^{2 n} } {\paren {2 n}!}
}}
{{eqn | r = 1 + \frac {x^2} 2 + \frac {5 x^4} {24} + \frac {61 x^6} {720} + \dfrac {1385 x^8} {40320} + \cdots
}}
{{end-eqn}}
where... | {{begin-eqn}}
{{eqn | l = \sec x
| r = \map \sech {i x}
| c = Secant in terms of Hyperbolic Secant
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {E_n \paren {i x}^n} {n!}
| c = {{Defof|Euler Numbers}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {i x}^{2 n} } {\paren {2 n}!}
... | The [[Definition:Real Secant Function|(real) secant function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \sec x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {E_{2 n} x^{2 n} } {\paren {2 n}!}
}}
{{eqn | r = 1 + \frac {x^2} 2 + \frac {5 x^4} {24} + \frac {61 x^... | {{begin-eqn}}
{{eqn | l = \sec x
| r = \map \sech {i x}
| c = [[Secant in terms of Hyperbolic Secant]]
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {E_n \paren {i x}^n} {n!}
| c = {{Defof|Euler Numbers}}
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \frac {E_{2 n} \paren {i x}^{2 n} } {\paren {2 n}!}... | Power Series Expansion for Secant Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Secant_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Secant_Function | [
"Power Series Expansion for Secant Function",
"Examples of Power Series",
"Euler Numbers",
"Secant Function"
] | [
"Definition:Secant Function/Real",
"Definition:Taylor Series",
"Definition:Euler Numbers",
"Definition:Convergent Series"
] | [
"Secant in terms of Hyperbolic Secant",
"Definition:Odd Integer"
] |
proofwiki-14258 | Power Series Expansion for Real Arccotangent Function | The arccotangent function has a Taylor series expansion:
:<nowiki>$\arccot x = \begin {cases} \ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \\
\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac 1 {\paren {2 n + 1} x^{2 n + 1} } & : x \ge 1 \\ \\
\ds \p... | From Sum of Arctangent and Arccotangent:
:$\arccot x = \dfrac \pi 2 - \arctan x$
The result follows from Power Series Expansion for Real Arctangent Function.
{{qed}} | The [[Definition:Inverse Cotangent|arccotangent function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
:<nowiki>$\arccot x = \begin {cases} \ds \frac \pi 2 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \\
\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \... | From [[Sum of Arctangent and Arccotangent]]:
:$\arccot x = \dfrac \pi 2 - \arctan x$
The result follows from [[Power Series Expansion for Real Arctangent Function]].
{{qed}} | Power Series Expansion for Real Arccotangent Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arccotangent_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arccotangent_Function | [
"Examples of Power Series",
"Arccotangent Function"
] | [
"Definition:Inverse Cotangent",
"Definition:Taylor Series"
] | [
"Sum of Arctangent and Arccotangent",
"Power Series Expansion for Real Arctangent Function"
] |
proofwiki-14259 | Power Series Expansion for Real Arcsecant Function | The arcsecant function has a Taylor Series expansion:
{{begin-eqn}}
{{eqn | l = \arcsec x
| r = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} x^{2 n + 1} }
| c =
}}
{{eqn | r = \frac \pi 2 - \paren {\frac 1 x + \frac 1 {2 \times 3 x^3} + \frac {1 \t... | From Arccosine of Reciprocal equals Arcsecant:
:$\arcsec x = \arccos \dfrac 1 x$
From Power Series Expansion for Real Arccosine Function:
:$\ds \arccos x = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}$
which is converges for $\size x \le 1$.
The ... | The [[Definition:Inverse Secant|arcsecant]] function has a [[Definition:Taylor Series|Taylor Series]] expansion:
{{begin-eqn}}
{{eqn | l = \arcsec x
| r = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} x^{2 n + 1} }
| c =
}}
{{eqn | r = \frac \pi 2 ... | From [[Arccosine of Reciprocal equals Arcsecant]]:
:$\arcsec x = \arccos \dfrac 1 x$
From [[Power Series Expansion for Real Arccosine Function]]:
:$\ds \arccos x = \frac \pi 2 - \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}$
which is converges for $\size x \l... | Power Series Expansion for Real Arcsecant Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arcsecant_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arcsecant_Function | [
"Examples of Power Series",
"Arcsecant Function"
] | [
"Definition:Inverse Secant",
"Definition:Taylor Series",
"Definition:Convergent Series"
] | [
"Arccosine of Reciprocal equals Arcsecant",
"Power Series Expansion for Real Arccosine Function"
] |
proofwiki-14260 | Power Series Expansion for Real Arccosecant Function | The arccosecant function has a Taylor Series expansion:
{{begin-eqn}}
{{eqn | l = \arccsc x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} x^{2 n + 1} }
| c =
}}
{{eqn | r = \frac 1 x + \frac 1 {2 \times 3 x^3} + \frac {1 \times 3} {2 \times 4 \times 5 x^5} ... | From Arcsine of Reciprocal equals Arccosecant:
:$\arccsc x = \arcsin \dfrac 1 x$
From Power Series Expansion for Real Arcsine Function:
:$\ds \arcsin x = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}$
which is converges for $\size x \le 1$.
The result follows b... | The [[Definition:Inverse Cosecant|arccosecant]] function has a [[Definition:Taylor Series|Taylor Series]] expansion:
{{begin-eqn}}
{{eqn | l = \arccsc x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n + 1} x^{2 n + 1} }
| c =
}}
{{eqn | r = \frac 1 x + \frac 1 {2... | From [[Arcsine of Reciprocal equals Arccosecant]]:
:$\arccsc x = \arcsin \dfrac 1 x$
From [[Power Series Expansion for Real Arcsine Function]]:
:$\ds \arcsin x = \sum_{n \mathop = 0}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2} \frac {x^{2 n + 1} } {2 n + 1}$
which is converges for $\size x \le 1$.
The resul... | Power Series Expansion for Real Arccosecant Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arccosecant_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Arccosecant_Function | [
"Examples of Power Series",
"Arccosecant Function"
] | [
"Definition:Inverse Cosecant",
"Definition:Taylor Series",
"Definition:Convergent Series"
] | [
"Arcsine of Reciprocal equals Arccosecant",
"Power Series Expansion for Real Arcsine Function"
] |
proofwiki-14261 | Power Series Expansion for Hyperbolic Sine Function | The hyperbolic sine function has the power series expansion:
{{begin-eqn}}
{{eqn | l = \sinh x
| r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} + \cdots
| c =
}}
{{end-eqn}}
valid for all $x \in... | From Derivative of Hyperbolic Sine:
:$\dfrac \d {\d x} \sinh x = \cosh x$
From Derivative of Hyperbolic Cosine:
:$\dfrac \d {\d x} \cosh x = \sinh x$
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac {\d^2} {\d x^2} \sinh x
| r = \sinh x
| c =
}}
{{end-eqn}}
and so for all $m \in \N$:
{{begin-eqn}}
{{eqn | ll= m = 2... | The [[Definition:Hyperbolic Sine|hyperbolic sine function]] has the [[Definition:Power Series|power series expansion]]:
{{begin-eqn}}
{{eqn | l = \sinh x
| r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c =
}}
{{eqn | r = x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7... | From [[Derivative of Hyperbolic Sine]]:
:$\dfrac \d {\d x} \sinh x = \cosh x$
From [[Derivative of Hyperbolic Cosine]]:
:$\dfrac \d {\d x} \cosh x = \sinh x$
Hence:
{{begin-eqn}}
{{eqn | l = \dfrac {\d^2} {\d x^2} \sinh x
| r = \sinh x
| c =
}}
{{end-eqn}}
and so for all $m \in \N$:
{{begin-eqn}}
{{e... | Power Series Expansion for Hyperbolic Sine Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Sine_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Sine_Function | [
"Hyperbolic Sine Function",
"Examples of Power Series",
"Taylor Series"
] | [
"Definition:Hyperbolic Sine",
"Definition:Power Series"
] | [
"Derivative of Hyperbolic Sine",
"Derivative of Hyperbolic Cosine",
"Definition:Maclaurin Series",
"Series of Power over Factorial Converges",
"Definition:Convergent Series"
] |
proofwiki-14262 | Power Series Expansion for Hyperbolic Cosine Function | The hyperbolic cosine function has the power series expansion:
{{begin-eqn}}
{{eqn | l = \cosh x
| r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots
| c =
}}
{{end-eqn}}
valid for all $x \in \R$. | From Derivative of Hyperbolic Cosine:
:$\dfrac \d {\d x} \cosh x = \sinh x$
From Derivative of Hyperbolic Sine:
:$\dfrac \d {\d x} \sinh x = \cosh x$
Hence:
:$\dfrac {\d^2} {\d x^2} \cosh x = \cosh x$
and so for all $m \in \N$:
{{begin-eqn}}
{{eqn | ll= m = 2 k:
| l = \dfrac {\d^m} {\d x^m} \cosh x
| r = \c... | The [[Definition:Hyperbolic Cosine|hyperbolic cosine function]] has the [[Definition:Power Series|power series expansion]]:
{{begin-eqn}}
{{eqn | l = \cosh x
| r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} +... | From [[Derivative of Hyperbolic Cosine]]:
:$\dfrac \d {\d x} \cosh x = \sinh x$
From [[Derivative of Hyperbolic Sine]]:
:$\dfrac \d {\d x} \sinh x = \cosh x$
Hence:
:$\dfrac {\d^2} {\d x^2} \cosh x = \cosh x$
and so for all $m \in \N$:
{{begin-eqn}}
{{eqn | ll= m = 2 k:
| l = \dfrac {\d^m} {\d x^m} \cosh x
... | Power Series Expansion for Hyperbolic Cosine Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Cosine_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Cosine_Function | [
"Hyperbolic Cosine Function",
"Examples of Power Series",
"Taylor Series"
] | [
"Definition:Hyperbolic Cosine",
"Definition:Power Series"
] | [
"Derivative of Hyperbolic Cosine",
"Derivative of Hyperbolic Sine",
"Definition:Maclaurin Series",
"Series of Power over Factorial Converges",
"Definition:Convergent Series"
] |
proofwiki-14263 | Power Series Expansion for Hyperbolic Tangent Function | The hyperbolic tangent function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \tanh x
| r = \sum_{n \mathop = 1}^\infty \frac {2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = x - \frac {x^3} 3 + \frac {2 x^5} {15} - \frac {17 x^7} {315} + \frac {62 x^9}... | From Power Series Expansion for Hyperbolic Cotangent Function:
:$(1): \quad \coth x = \ds \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}$
Then:
{{begin-eqn}}
{{eqn | l = \tanh x
| r = 2 \coth 2 x - \coth x
| c = Sum of Hyperbolic Tangent and Cotangent
}}
{{eqn | r = 2 \s... | The [[Definition:Hyperbolic Tangent|hyperbolic tangent function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \tanh x
| r = \sum_{n \mathop = 1}^\infty \frac {2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = x - \frac {x^3... | From [[Power Series Expansion for Hyperbolic Cotangent Function]]:
:$(1): \quad \coth x = \ds \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}$
Then:
{{begin-eqn}}
{{eqn | l = \tanh x
| r = 2 \coth 2 x - \coth x
| c = [[Sum of Hyperbolic Tangent and Cotangent]]
}}
{{eqn... | Power Series Expansion for Hyperbolic Tangent Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Tangent_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Tangent_Function | [
"Power Series Expansion for Hyperbolic Tangent Function",
"Examples of Power Series",
"Bernoulli Numbers",
"Hyperbolic Tangent Function"
] | [
"Definition:Hyperbolic Tangent",
"Definition:Taylor Series",
"Definition:Bernoulli Numbers",
"Definition:Convergent Series"
] | [
"Power Series Expansion for Hyperbolic Cotangent Function",
"Sum of Hyperbolic Tangent and Cotangent",
"Combination Theorem for Limits of Functions/Real",
"Asymptotic Formula for Bernoulli Numbers",
"Ratio Test",
"Definition:Convergent Series"
] |
proofwiki-14264 | Power Series Expansion for Hyperbolic Cotangent Function | The hyperbolic cotangent function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \coth x
| r = \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 x + \frac x 3 - \frac {x^3} {45} + \frac {2 x^5} {945} - \frac {x^7} {4725} + \cdots
... | {{begin-eqn}}
{{eqn | l = \coth x
| r = \frac {e^x + e^{-x} } {e^x - e^{-x} }
| c = {{Defof|Hyperbolic Cotangent|index = 1}}
}}
{{eqn | r = \frac {e^{2 x} + 1} {e^{2 x} - 1}
| c =
}}
{{eqn | r = 1 + \frac 2 {e^{2 x} - 1}
| c =
}}
{{eqn | r = 1 + \frac 1 x \frac {2 x} {e^{2 x} - 1}
| c =
}}... | The [[Definition:Hyperbolic Cotangent|hyperbolic cotangent function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \coth x
| r = \sum_{n \mathop = 0}^\infty \frac {2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 x + \frac x 3 - \frac {... | {{begin-eqn}}
{{eqn | l = \coth x
| r = \frac {e^x + e^{-x} } {e^x - e^{-x} }
| c = {{Defof|Hyperbolic Cotangent|index = 1}}
}}
{{eqn | r = \frac {e^{2 x} + 1} {e^{2 x} - 1}
| c =
}}
{{eqn | r = 1 + \frac 2 {e^{2 x} - 1}
| c =
}}
{{eqn | r = 1 + \frac 1 x \frac {2 x} {e^{2 x} - 1}
| c =
}}... | Power Series Expansion for Hyperbolic Cotangent Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Cotangent_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Cotangent_Function | [
"Power Series Expansion for Hyperbolic Cotangent Function",
"Examples of Power Series",
"Bernoulli Numbers",
"Hyperbolic Cotangent Function"
] | [
"Definition:Hyperbolic Cotangent",
"Definition:Taylor Series",
"Definition:Bernoulli Numbers",
"Definition:Convergent Series"
] | [
"Odd Bernoulli Numbers Vanish",
"Combination Theorem for Limits of Functions/Real",
"Asymptotic Formula for Bernoulli Numbers",
"Ratio Test",
"Definition:Convergent Series"
] |
proofwiki-14265 | Power Series Expansion for Hyperbolic Secant Function | The hyperbolic secant function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \sech x
| r = \sum_{n \mathop = 0}^\infty \frac {E_{2 n} x^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = 1 - \frac {x^2} 2 + \frac {5 x^4} {24} - \frac {61 x^6} {720} + \cdots
| c =
}}
{{end-eqn}}
where $E_{2 n}$... | By definition of the Euler numbers:
:$\ds \sech x = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}$
From Odd Euler Numbers Vanish:
:$E_{2 k + 1} = 0$
for $k \in \Z$.
Hence the result.
{{qed}} | The [[Definition:Hyperbolic Secant|hyperbolic secant function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \sech x
| r = \sum_{n \mathop = 0}^\infty \frac {E_{2 n} x^{2 n} } {\paren {2 n}!}
| c =
}}
{{eqn | r = 1 - \frac {x^2} 2 + \frac {5 x^4} {24} - \frac {61 ... | By definition of the [[Definition:Euler Numbers|Euler numbers]]:
:$\ds \sech x = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}$
From [[Odd Euler Numbers Vanish]]:
:$E_{2 k + 1} = 0$
for $k \in \Z$.
Hence the result.
{{qed}} | Power Series Expansion for Hyperbolic Secant Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Secant_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Secant_Function | [
"Power Series Expansion for Hyperbolic Secant Function",
"Examples of Power Series",
"Euler Numbers",
"Hyperbolic Secant Function"
] | [
"Definition:Hyperbolic Secant",
"Definition:Taylor Series",
"Definition:Euler Numbers",
"Definition:Convergent Series"
] | [
"Definition:Euler Numbers",
"Odd Euler Numbers Vanish"
] |
proofwiki-14266 | Power Series Expansion for Hyperbolic Cosecant Function | The hyperbolic cosecant function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \csch x
| r = \sum_{n \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 x - \frac x 6 + \frac {7 x^3} {360} - \frac {31 x^5} {15 \, 120} + \c... | {{begin-eqn}}
{{eqn | l = \sinh x
| r = 2 \sinh \dfrac x 2 \cosh \dfrac x 2
| c = Double Angle Formula for Hyperbolic Sine
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac 1 {\sinh x}
| r = \dfrac 1 {2 \sinh \dfrac x 2 \cosh \dfrac x 2}
| c = taking the reciprocal of both sides
}}
{{eqn | ll= \... | The [[Definition:Hyperbolic Cosecant|hyperbolic cosecant function]] has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \csch x
| r = \sum_{n \mathop = 0}^\infty \dfrac {2 \paren {1 - 2^{2 n - 1} } B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 x - ... | {{begin-eqn}}
{{eqn | l = \sinh x
| r = 2 \sinh \dfrac x 2 \cosh \dfrac x 2
| c = [[Double Angle Formula for Hyperbolic Sine]]
}}
{{eqn | ll= \leadstoandfrom
| l = \dfrac 1 {\sinh x}
| r = \dfrac 1 {2 \sinh \dfrac x 2 \cosh \dfrac x 2}
| c = taking the reciprocal of both sides
}}
{{eqn | l... | Power Series Expansion for Hyperbolic Cosecant Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Cosecant_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Hyperbolic_Cosecant_Function | [
"Power Series Expansion for Hyperbolic Cosecant Function",
"Examples of Power Series",
"Bernoulli Numbers",
"Hyperbolic Cosecant Function"
] | [
"Definition:Hyperbolic Cosecant",
"Definition:Taylor Series",
"Definition:Bernoulli Numbers",
"Definition:Convergent Series"
] | [
"Double Angle Formulas/Hyperbolic Sine",
"Cosecant is Reciprocal of Sine",
"Secant is Reciprocal of Cosine",
"Odd Bernoulli Numbers Vanish"
] |
proofwiki-14267 | Sum of Hyperbolic Tangent and Cotangent | :$\tanh x + \coth x = 2 \coth 2 x$ | {{begin-eqn}}
{{eqn | l = \tanh x + \coth x
| r = \frac {\sinh x} {\cosh x} + \frac {\cosh x} {\sinh x}
| c = {{Defof|Hyperbolic Tangent}} and Hyperbolic Cotangent
}}
{{eqn | r = \frac {\cosh^2 x + \sinh^2 x} {\sinh x \cosh x}
| c =
}}
{{eqn | r = 2 \frac {\cosh^2 x + \sinh^2 x} {2 \sinh x \cosh x}
... | :$\tanh x + \coth x = 2 \coth 2 x$ | {{begin-eqn}}
{{eqn | l = \tanh x + \coth x
| r = \frac {\sinh x} {\cosh x} + \frac {\cosh x} {\sinh x}
| c = {{Defof|Hyperbolic Tangent}} and [[Definition:Hyperbolic Cotangent|Hyperbolic Cotangent]]
}}
{{eqn | r = \frac {\cosh^2 x + \sinh^2 x} {\sinh x \cosh x}
| c =
}}
{{eqn | r = 2 \frac {\cosh^2 ... | Sum of Hyperbolic Tangent and Cotangent | https://proofwiki.org/wiki/Sum_of_Hyperbolic_Tangent_and_Cotangent | https://proofwiki.org/wiki/Sum_of_Hyperbolic_Tangent_and_Cotangent | [
"Hyperbolic Tangent Function",
"Hyperbolic Cotangent Function"
] | [] | [
"Definition:Hyperbolic Cotangent",
"Double Angle Formulas/Hyperbolic Sine",
"Double Angle Formulas/Hyperbolic Cosine",
"Category:Hyperbolic Tangent Function",
"Category:Hyperbolic Cotangent Function"
] |
proofwiki-14268 | Power Series Expansion for Real Area Hyperbolic Cosine | The (real) area hyperbolic cosine function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \arcosh x
| r = \map \ln {2 x} - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } }
| c =
}}
{{eqn | r = \map \ln {2 x} - \paren {\dfrac 1 {2 \times 2 x... | === Lemma 1 ===
{{:Power Series Expansion for Real Area Hyperbolic Cosine/Lemma 1}}{{qed|lemma}}
We have that the (real) area hyperbolic cosine is defined for $x \ge 1$.
Let $z = \dfrac 1 x$.
Then we have:
:$0 < \dfrac 1 z \le 1$
Now we consider:
{{begin-eqn}}
{{eqn | l = \map \arcosh {\dfrac 1 z} + \map \ln {2 z}
... | The [[Definition:Real Area Hyperbolic Cosine|(real) area hyperbolic cosine]] function has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \arcosh x
| r = \map \ln {2 x} - \paren {\sum_{n \mathop = 1}^\infty \frac {\paren {2 n}!} {2^{2 n} \paren {n!}^2 \paren {2 n} x^{2 n} } }
... | === [[Power Series Expansion for Real Area Hyperbolic Cosine/Lemma 1|Lemma 1]] ===
{{:Power Series Expansion for Real Area Hyperbolic Cosine/Lemma 1}}{{qed|lemma}}
We have that the [[Definition:Real Area Hyperbolic Cosine|(real) area hyperbolic cosine]] is defined for $x \ge 1$.
Let $z = \dfrac 1 x$.
Then we have:
... | Power Series Expansion for Real Area Hyperbolic Cosine | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Cosine | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Cosine | [
"Power Series Expansion for Real Area Hyperbolic Cosine",
"Examples of Power Series",
"Inverse Hyperbolic Cosine"
] | [
"Definition:Inverse Hyperbolic Cosine/Real/Principal Branch",
"Definition:Taylor Series"
] | [
"Power Series Expansion for Real Area Hyperbolic Cosine/Lemma 1",
"Definition:Inverse Hyperbolic Cosine/Real/Principal Branch",
"Sum of Logarithms",
"Sum of Logarithms",
"Logarithm of Reciprocal",
"Sum of Logarithms",
"Power Series Expansion for Real Area Hyperbolic Cosine/Lemma 2",
"Sum of Logarithms... |
proofwiki-14269 | Power Series Expansion for Real Area Hyperbolic Tangent | The (real) area hyperbolic tangent function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \artanh x
| r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}
| c =
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots
| c =
}}
{{end-eqn}}
for $\size x < 1$. | From Sum of Infinite Geometric Sequence:
:$(1): \quad \ds \frac 1 {1 - x^2} = \sum_{n \mathop = 0}^\infty \paren {x^2}^n$
for $-1 < x < 1$.
From Power Series is Termwise Integrable within Radius of Convergence, $(1)$ can be integrated term by term:
{{begin-eqn}}
{{eqn | l = \int_0^x \frac 1 {1 - t^2} \rd t
| r = ... | The [[Definition:Real Area Hyperbolic Tangent|(real) area hyperbolic tangent]] function has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \artanh x
| r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}
| c =
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {x^5} 5 + ... | From [[Sum of Infinite Geometric Sequence]]:
:$(1): \quad \ds \frac 1 {1 - x^2} = \sum_{n \mathop = 0}^\infty \paren {x^2}^n$
for $-1 < x < 1$.
From [[Power Series is Termwise Integrable within Radius of Convergence]], $(1)$ can be [[Definition:Integration|integrated]] term by term:
{{begin-eqn}}
{{eqn | l = \int_0^x... | Power Series Expansion for Real Area Hyperbolic Tangent | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Tangent | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Tangent | [
"Examples of Power Series",
"Inverse Hyperbolic Tangent"
] | [
"Definition:Inverse Hyperbolic Tangent/Real/Definition 2",
"Definition:Taylor Series"
] | [
"Sum of Infinite Geometric Sequence",
"Power Series is Termwise Integrable within Radius of Convergence",
"Definition:Primitive (Calculus)/Integration",
"Primitive of Reciprocal of x squared plus a squared/Arctangent Form",
"Integral of Power",
"Definition:Real Sequence",
"Definition:Decreasing/Sequence... |
proofwiki-14270 | Power Series Expansion for Real Area Hyperbolic Cotangent | The (real) area hyperbolic cotangent function has a Taylor series expansion:
{{begin-eqn}}
{{eqn | l = \arcoth x
| r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1} x^{2 n + 1} }
| c =
}}
{{eqn | r = \frac 1 x + \frac 1 {3 x^3} + \frac 1 {5 x^5} + \frac 1 {7 x^7} + \cdots
| c =
}}
{{end-eqn}... | From Power Series Expansion for Real Area Hyperbolic Tangent:
{{begin-eqn}}
{{eqn | l = \artanh x
| r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}
| c =
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots
| c =
}}
{{end-eqn}}
for $\size x < 1$.
From Real Area Hy... | The [[Definition:Real Area Hyperbolic Cotangent|(real) area hyperbolic cotangent]] function has a [[Definition:Taylor Series|Taylor series expansion]]:
{{begin-eqn}}
{{eqn | l = \arcoth x
| r = \sum_{n \mathop = 0}^\infty \frac 1 {\paren {2 n + 1} x^{2 n + 1} }
| c =
}}
{{eqn | r = \frac 1 x + \frac 1 {3 ... | From [[Power Series Expansion for Real Area Hyperbolic Tangent]]:
{{begin-eqn}}
{{eqn | l = \artanh x
| r = \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {2 n + 1}
| c =
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {x^5} 5 + \frac {x^7} 7 + \cdots
| c =
}}
{{end-eqn}}
for $\size x < 1$.
From [[Rea... | Power Series Expansion for Real Area Hyperbolic Cotangent | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Cotangent | https://proofwiki.org/wiki/Power_Series_Expansion_for_Real_Area_Hyperbolic_Cotangent | [
"Examples of Power Series",
"Inverse Hyperbolic Cotangent"
] | [
"Definition:Inverse Hyperbolic Cotangent/Real/Definition 2",
"Definition:Taylor Series"
] | [
"Power Series Expansion for Real Area Hyperbolic Tangent",
"Real Area Hyperbolic Tangent of Reciprocal equals Real Area Hyperbolic Cotangent"
] |
proofwiki-14271 | Power Series Expansion for Exponential of Sine of x | :$e^{\sin x} = 1 + x + \dfrac {x^2} 2 - \dfrac {x^4} 8 - \dfrac {x^5} {15} + \cdots$
for all $x \in \R$. | Let $\map f x = e^{\sin x}$.
Then:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \map f x
| r = \cos x \, e^{\sin x}
| c = Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2} {\d x^2} \map f x
| r = \cos x \frac \d {\d x} e^{\sin x} + e^{\sin x} \frac \d {\d x} \cos x
| c = P... | :$e^{\sin x} = 1 + x + \dfrac {x^2} 2 - \dfrac {x^4} 8 - \dfrac {x^5} {15} + \cdots$
for all $x \in \R$. | Let $\map f x = e^{\sin x}$.
Then:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \map f x
| r = \cos x \, e^{\sin x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2} {\d x^2} \map f x
| r = \cos x \frac \d {\d x} e^{\sin x} + e^{\sin x} \frac \d {\d x} \cos x
|... | Power Series Expansion for Exponential of Sine of x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_Sine_of_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_Sine_of_x | [
"Examples of Power Series",
"Sine Function",
"Exponential Function"
] | [] | [
"Derivative of Composite Function",
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Definition:Taylor Series",
"Sine of Zero is Zero",
"Exponential of Zero",
"Cosine of Zero is One"
] |
proofwiki-14272 | Power Series Expansion for Exponential of Cosine of x | {{begin-eqn}}
{{eqn | l = e^{\cos x}
| r = e \paren {e^{\cos x - 1} }
| c =
}}
{{eqn | r = e \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^m \map {P_2} {2 m} } {2 m!} x^{2 m} }
| c = where $\map {P_2} {2 m}$ is the partition of the set of size $2 m$ into even blocks
}}
{{eqn | r = e \paren {... | Let $\map f x = e^{\cos x}$.
Then:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \map f x
| r = -\sin x \, e^{\cos x}
| c = Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2} {\d x^2} \map f x
| r = -\sin x \frac \d {\d x} e^{\cos x} + e^{\cos x} \map {\frac \d {\d x} } {-\sin x}
... | {{begin-eqn}}
{{eqn | l = e^{\cos x}
| r = e \paren {e^{\cos x - 1} }
| c =
}}
{{eqn | r = e \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^m \map {P_2} {2 m} } {2 m!} x^{2 m} }
| c = where $\map {P_2} {2 m}$ is the partition of the set of size $2 m$ into even blocks
}}
{{eqn | r = e \paren ... | Let $\map f x = e^{\cos x}$.
Then:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \map f x
| r = -\sin x \, e^{\cos x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2} {\d x^2} \map f x
| r = -\sin x \frac \d {\d x} e^{\cos x} + e^{\cos x} \map {\frac \d {\d x} } {-\s... | Power Series Expansion for Exponential of Cosine of x/Proof 1 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_Cosine_of_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_Cosine_of_x/Proof_1 | [
"Power Series Expansion for Exponential of Cosine of x",
"Examples of Power Series",
"Cosine Function",
"Exponential Function"
] | [] | [
"Derivative of Composite Function",
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Definition:Taylor Series",
"Sine of Zero is Zero",
"Exponential of Zero",
"Cosine of Zero is One"
] |
proofwiki-14273 | Power Series Expansion for Exponential of Cosine of x | {{begin-eqn}}
{{eqn | l = e^{\cos x}
| r = e \paren {e^{\cos x - 1} }
| c =
}}
{{eqn | r = e \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^m \map {P_2} {2 m} } {2 m!} x^{2 m} }
| c = where $\map {P_2} {2 m}$ is the partition of the set of size $2 m$ into even blocks
}}
{{eqn | r = e \paren {... | A result in combinatorics known as the Exponential formula states that if
:$\ds \map f x = \sum_{n \mathop = 1}^\infty \frac {c_n} {n!} x^n $
then:
:$\ds e^{\map f x} = \sum_{k \mathop = 0}^\infty \frac {\map {B_k} {c_1, c_2, \ldots, c_k} } {k!} x^k$
where $\map {B_k} {c_1, c_2, \ldots, c_k}$ is the $k$th complete Bel... | {{begin-eqn}}
{{eqn | l = e^{\cos x}
| r = e \paren {e^{\cos x - 1} }
| c =
}}
{{eqn | r = e \paren {\sum_{n \mathop = 0}^\infty \frac {\paren {-1}^m \map {P_2} {2 m} } {2 m!} x^{2 m} }
| c = where $\map {P_2} {2 m}$ is the partition of the set of size $2 m$ into even blocks
}}
{{eqn | r = e \paren ... | A result in combinatorics known as the Exponential formula states that if
:$\ds \map f x = \sum_{n \mathop = 1}^\infty \frac {c_n} {n!} x^n $
then:
:$\ds e^{\map f x} = \sum_{k \mathop = 0}^\infty \frac {\map {B_k} {c_1, c_2, \ldots, c_k} } {k!} x^k$
where $\map {B_k} {c_1, c_2, \ldots, c_k}$ is the $k$th complete Bel... | Power Series Expansion for Exponential of Cosine of x/Proof 2 | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_Cosine_of_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_Cosine_of_x/Proof_2 | [
"Power Series Expansion for Exponential of Cosine of x",
"Examples of Power Series",
"Cosine Function",
"Exponential Function"
] | [] | [
"Power Series Expansion for Cosine Function"
] |
proofwiki-14274 | Power Series Expansion for Exponential of Tangent of x | :$e^{\tan x} = 1 + x + \dfrac {x^2} 2 + \dfrac {x^3} 2 + \dfrac {3 x^4} 8 + \cdots$
for all $x \in \R$ such that $\size x < \dfrac \pi 2$. | Let $\map f x = e^{\tan x}$.
Then:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \map f x
| r = \sec^2 x \, e^{\tan x}
| c = Chain Rule for Derivatives
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2} {\d x^2} \map f x
| r = \sec^2 x \frac \d {\d x} e^{\tan x} + e^{\tan x} \frac \d {\d x} \sec^2 x
|... | :$e^{\tan x} = 1 + x + \dfrac {x^2} 2 + \dfrac {x^3} 2 + \dfrac {3 x^4} 8 + \cdots$
for all $x \in \R$ such that $\size x < \dfrac \pi 2$. | Let $\map f x = e^{\tan x}$.
Then:
{{begin-eqn}}
{{eqn | l = \frac \d {\d x} \map f x
| r = \sec^2 x \, e^{\tan x}
| c = [[Chain Rule for Derivatives]]
}}
{{eqn | ll= \leadsto
| l = \frac {\d^2} {\d x^2} \map f x
| r = \sec^2 x \frac \d {\d x} e^{\tan x} + e^{\tan x} \frac \d {\d x} \sec^2 x
... | Power Series Expansion for Exponential of Tangent of x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_Tangent_of_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_Tangent_of_x | [
"Examples of Power Series",
"Tangent Function",
"Exponential Function"
] | [] | [
"Derivative of Composite Function",
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Product Rule for Derivatives",
"Definition:Taylor Series"
] |
proofwiki-14275 | Power Series Expansion for Exponential of x by Sine of x | {{begin-eqn}}
{{eqn | l = e^x \sin x
| r = \sum_{n \mathop = 1}^\infty \frac {2^{n / 2} \, \map \sin {n \pi / 4} x^n} {n!}
| c =
}}
{{eqn | r = x + x^2 + \frac {x^3} 3 - \frac {x^5} {30} - \frac {x^6} {90} + \cdots
| c =
}}
{{end-eqn}}
for all $x \in \R$. | Let $\map f x = e^x \sin x$.
By definition of Maclaurin series:
:$(1): \quad \map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$
where $\map {f^{\paren n} } 0$ denotes the $n$th derivative of $f$ {{WRT|Differentiation}} $x$ evaluated at $x = 0$.
It remains to be shown that:
:$\map {f^... | {{begin-eqn}}
{{eqn | l = e^x \sin x
| r = \sum_{n \mathop = 1}^\infty \frac {2^{n / 2} \, \map \sin {n \pi / 4} x^n} {n!}
| c =
}}
{{eqn | r = x + x^2 + \frac {x^3} 3 - \frac {x^5} {30} - \frac {x^6} {90} + \cdots
| c =
}}
{{end-eqn}}
for all $x \in \R$. | Let $\map f x = e^x \sin x$.
By definition of [[Definition:Maclaurin Series|Maclaurin series]]:
:$(1): \quad \map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$
where $\map {f^{\paren n} } 0$ denotes the [[Definition:Higher Derivative|$n$th derivative]] of $f$ {{WRT|Differentiation... | Power Series Expansion for Exponential of x by Sine of x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_x_by_Sine_of_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_x_by_Sine_of_x | [
"Examples of Power Series",
"Sine Function",
"Exponential Function"
] | [] | [
"Definition:Maclaurin Series",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-14276 | Sine of x plus Cosine of x/Sine Form | :$\sin x + \cos x = \sqrt 2 \sin \left({x + \dfrac \pi 4}\right)$ | {{begin-eqn}}
{{eqn | l = \sin x + \cos x
| r = \sqrt 2 \cos \left({x - \dfrac \pi 4}\right)
| c = Sine of x plus Cosine of x: Cosine Form
}}
{{eqn | r = \sqrt 2 \sin \left({x - \frac \pi 4 + \frac \pi 4}\right)
| c = Sine of Angle plus Right Angle
}}
{{eqn | r = \sqrt 2 \sin \left({x + \frac \pi 4}\r... | :$\sin x + \cos x = \sqrt 2 \sin \left({x + \dfrac \pi 4}\right)$ | {{begin-eqn}}
{{eqn | l = \sin x + \cos x
| r = \sqrt 2 \cos \left({x - \dfrac \pi 4}\right)
| c = [[Sine of x plus Cosine of x/Cosine Form|Sine of x plus Cosine of x: Cosine Form]]
}}
{{eqn | r = \sqrt 2 \sin \left({x - \frac \pi 4 + \frac \pi 4}\right)
| c = [[Sine of Angle plus Right Angle]]
}}
{{e... | Sine of x plus Cosine of x/Sine Form | https://proofwiki.org/wiki/Sine_of_x_plus_Cosine_of_x/Sine_Form | https://proofwiki.org/wiki/Sine_of_x_plus_Cosine_of_x/Sine_Form | [
"Sine Function",
"Cosine Function"
] | [] | [
"Sine of x plus Cosine of x/Cosine Form",
"Sine of Angle plus Right Angle",
"Category:Sine Function",
"Category:Cosine Function"
] |
proofwiki-14277 | Cosine of x minus Sine of x/Sine Form | :$\cos x - \sin x = \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}$ | {{begin-eqn}}
{{eqn | l = \cos x - \sin x
| r = \cos x - \map \cos {\frac \pi 2 - x}
| c = Cosine of Complement equals Sine
}}
{{eqn | r = -2 \, \map \sin {\frac {x + \paren {\frac \pi 2 - x} } 2} \map \sin {\frac {x - \paren {\frac \pi 2 - x} } 2}
| c = Cosine minus Cosine
}}
{{eqn | r = -2 \sin \fra... | :$\cos x - \sin x = \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}$ | {{begin-eqn}}
{{eqn | l = \cos x - \sin x
| r = \cos x - \map \cos {\frac \pi 2 - x}
| c = [[Cosine of Complement equals Sine]]
}}
{{eqn | r = -2 \, \map \sin {\frac {x + \paren {\frac \pi 2 - x} } 2} \map \sin {\frac {x - \paren {\frac \pi 2 - x} } 2}
| c = [[Cosine minus Cosine]]
}}
{{eqn | r = -2 \... | Cosine of x minus Sine of x/Sine Form | https://proofwiki.org/wiki/Cosine_of_x_minus_Sine_of_x/Sine_Form | https://proofwiki.org/wiki/Cosine_of_x_minus_Sine_of_x/Sine_Form | [
"Cosine of x minus Sine of x"
] | [] | [
"Cosine of Complement equals Sine",
"Prosthaphaeresis Formulas/Cosine minus Cosine",
"Sine of 45 Degrees",
"Sine of Angle plus Straight Angle",
"Category:Cosine of x minus Sine of x"
] |
proofwiki-14278 | Cosine of x minus Sine of x/Cosine Form | :$\cos x - \sin x = \sqrt 2 \, \map \cos {x + \dfrac \pi 4}$ | {{begin-eqn}}
{{eqn | l = \cos x - \sin x
| r = \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}
| c = Cosine of x minus Sine of x: Sine Form
}}
{{eqn | r = \sqrt 2 \, \map \cos {\frac \pi 2 - \paren {x + \dfrac {3 \pi} 4} }
| c = Cosine of Complement equals Sine
}}
{{eqn | r = \sqrt 2 \, \map \cos {\frac ... | :$\cos x - \sin x = \sqrt 2 \, \map \cos {x + \dfrac \pi 4}$ | {{begin-eqn}}
{{eqn | l = \cos x - \sin x
| r = \sqrt 2 \, \map \sin {x + \dfrac {3 \pi} 4}
| c = [[Cosine of x minus Sine of x/Sine Form|Cosine of x minus Sine of x: Sine Form]]
}}
{{eqn | r = \sqrt 2 \, \map \cos {\frac \pi 2 - \paren {x + \dfrac {3 \pi} 4} }
| c = [[Cosine of Complement equals Sine... | Cosine of x minus Sine of x/Cosine Form | https://proofwiki.org/wiki/Cosine_of_x_minus_Sine_of_x/Cosine_Form | https://proofwiki.org/wiki/Cosine_of_x_minus_Sine_of_x/Cosine_Form | [
"Cosine of x minus Sine of x"
] | [] | [
"Cosine of x minus Sine of x/Sine Form",
"Cosine of Complement equals Sine",
"Cosine Function is Even",
"Category:Cosine of x minus Sine of x"
] |
proofwiki-14279 | Power Series Expansion for Exponential of x by Cosine of x | {{begin-eqn}}
{{eqn | l = e^x \cos x
| r = \sum_{n \mathop = 1}^\infty \frac {2^{n / 2} \, \map \cos {n \pi / 4} x^n} {n!}
| c =
}}
{{eqn | r = 1 + x - \frac {x^3} 3 - \frac {x^4} 6 + \cdots
| c =
}}
{{end-eqn}}
for all $x \in \R$. | Let $\map f x = e^x \cos x$.
By definition of Maclaurin series:
:$(1): \quad \map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$
where $\map {f^{\paren n} } 0$ denotes the $n$th derivative of $f$ {{WRT|Differentiation}} $x$ evaluated at $x = 0$.
It remains to be shown that:
:$\map {f^... | {{begin-eqn}}
{{eqn | l = e^x \cos x
| r = \sum_{n \mathop = 1}^\infty \frac {2^{n / 2} \, \map \cos {n \pi / 4} x^n} {n!}
| c =
}}
{{eqn | r = 1 + x - \frac {x^3} 3 - \frac {x^4} 6 + \cdots
| c =
}}
{{end-eqn}}
for all $x \in \R$. | Let $\map f x = e^x \cos x$.
By definition of [[Definition:Maclaurin Series|Maclaurin series]]:
:$(1): \quad \map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$
where $\map {f^{\paren n} } 0$ denotes the [[Definition:Higher Derivative|$n$th derivative]] of $f$ {{WRT|Differentiation... | Power Series Expansion for Exponential of x by Cosine of x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_x_by_Cosine_of_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Exponential_of_x_by_Cosine_of_x | [
"Examples of Power Series",
"Cosine Function",
"Exponential Function"
] | [] | [
"Definition:Maclaurin Series",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-14280 | Power Series Expansion for Logarithm of Sine of x | {{begin-eqn}}
{{eqn | l = \ln \size {\sin x}
| r = \ln \size x - \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n 2^{2 n - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}
| c =
}}
{{eqn | r = \ln \size x - \frac {x^2} 6 - \frac {x^4} {180} - \frac {x^6} {2835} + \cdots
| c =
}}
{{end-eqn}}
for all $x \in ... | From Power Series Expansion for Cotangent Function:
{{begin-eqn}}
{{eqn | n = 1
| l = \cot x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 x - \frac x 3 - \frac {x^3} {45} - \frac {2 x^5} {945} + \cdots
... | {{begin-eqn}}
{{eqn | l = \ln \size {\sin x}
| r = \ln \size x - \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n 2^{2 n - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}
| c =
}}
{{eqn | r = \ln \size x - \frac {x^2} 6 - \frac {x^4} {180} - \frac {x^6} {2835} + \cdots
| c =
}}
{{end-eqn}}
for all $x \i... | From [[Power Series Expansion for Cotangent Function]]:
{{begin-eqn}}
{{eqn | n = 1
| l = \cot x
| r = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = \frac 1 x - \frac x 3 - \frac {x^3} {45} - \frac {2 x^5} {945} + \cdots
... | Power Series Expansion for Logarithm of Sine of x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_Sine_of_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_Sine_of_x | [
"Power Series Expansion for Logarithm of Sine of x",
"Examples of Power Series",
"Logarithms",
"Sine Function"
] | [] | [
"Power Series Expansion for Cotangent Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Definition:Primitive (Calculus)/Integration",
"Definition:Zeroth",
"Primitive of Cotangent Function",
"Integral of Power",
"Primitive of Reciprocal"
] |
proofwiki-14281 | Power Series Expansion for Logarithm of Cosine of x | {{begin-eqn}}
{{eqn | l = \ln \size {\cos x}
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n 2^{2 n - 1} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}
| c =
}}
{{eqn | r = -\frac {x^2} 2 - \frac {x^4} {12} - \frac {x^6} {45} - \frac {17 x^8} {2520} - \cdots
| c =
}}
{{end-eqn}}
... | From Power Series Expansion for Tangent Function:
{{begin-eqn}}
{{eqn | n = 1
| l = \tan x
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {2 x^5} {15} + \frac {17 x^7} {3... | {{begin-eqn}}
{{eqn | l = \ln \size {\cos x}
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n 2^{2 n - 1} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}
| c =
}}
{{eqn | r = -\frac {x^2} 2 - \frac {x^4} {12} - \frac {x^6} {45} - \frac {17 x^8} {2520} - \cdots
| c =
}}
{{end-eqn}}
... | From [[Power Series Expansion for Tangent Function]]:
{{begin-eqn}}
{{eqn | n = 1
| l = \tan x
| r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n} - 1} B_{2 n} \, x^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | r = x + \frac {x^3} 3 + \frac {2 x^5} {15} + \frac {17 x^7... | Power Series Expansion for Logarithm of Cosine of x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_Cosine_of_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_Cosine_of_x | [
"Power Series Expansion for Logarithm of Cosine of x",
"Examples of Power Series",
"Logarithms",
"Cosine Function"
] | [] | [
"Power Series Expansion for Tangent Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Definition:Primitive (Calculus)/Integration",
"Primitive of Tangent Function/Cosine Form",
"Integral of Power"
] |
proofwiki-14282 | Expectation of Normal Distribution | Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
Then:
:$\expect X = \mu$ | From the definition of the normal distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt{2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
From the definition of the expected value of a continuous random variable:
:$\ds \expect X = \int_{-\infty}^\infty x \map {f_X} ... | Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the [[Definition:Normal Distribution|normal distribution]].
Then:
:$\expect X = \mu$ | From the definition of the [[Definition:Normal Distribution|normal distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt{2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
From the definition of the [[Definition:Expectat... | Expectation of Normal Distribution/Proof 1 | https://proofwiki.org/wiki/Expectation_of_Normal_Distribution | https://proofwiki.org/wiki/Expectation_of_Normal_Distribution/Proof_1 | [
"Expectation of Normal Distribution",
"Normal Distribution",
"Expectation"
] | [
"Definition:Normal Distribution"
] | [
"Definition:Normal Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Integration by Substitution",
"Fundamental Theorem of Calculus",
"Gaussian Integral",
"Exponential Tends to Zero and Infinity"
] |
proofwiki-14283 | Expectation of Normal Distribution | Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
Then:
:$\expect X = \mu$ | By Moment Generating Function of Normal Distribution, the moment generating function of $X$ is given by:
:$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$
From Moment in terms of Moment Generating Function:
:$\expect X = \map {M'_X} 0$
We have:
{{begin-eqn}}
{{eqn | l = \map {M_X'} t
| r = \frac \d... | Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the [[Definition:Normal Distribution|normal distribution]].
Then:
:$\expect X = \mu$ | By [[Moment Generating Function of Normal Distribution]], the [[Definition:Moment Generating Function|moment generating function]] of $X$ is given by:
:$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$
From [[Moment in terms of Moment Generating Function]]:
:$\expect X = \map {M'_X} 0$
We have:
{{beg... | Expectation of Normal Distribution/Proof 2 | https://proofwiki.org/wiki/Expectation_of_Normal_Distribution | https://proofwiki.org/wiki/Expectation_of_Normal_Distribution/Proof_2 | [
"Expectation of Normal Distribution",
"Normal Distribution",
"Expectation"
] | [
"Definition:Normal Distribution"
] | [
"Moment Generating Function of Normal Distribution",
"Definition:Moment Generating Function",
"Moment in terms of Moment Generating Function",
"Derivative of Composite Function",
"Power Rule for Derivatives",
"Derivative of Exponential Function",
"Exponential of Zero"
] |
proofwiki-14284 | Power Series Expansion for Logarithm of Tangent of x | {{begin-eqn}}
{{eqn | l = \ln \size {\tan x}
| r = \ln \size x + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n - 1} - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}
| c =
}}
{{eqn | r = \ln \size x + \frac {x^2} 3 + \frac {7 x^4} {90} + \frac {62 x^6} {2835} + \cdots
| c = ... | From Power Series Expansion for Cosecant Function:
{{begin-eqn}}
{{eqn | l = 2 \csc 2 x
| r = 2 \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {2 x}^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = 2 \csc 2 x
| r = ... | {{begin-eqn}}
{{eqn | l = \ln \size {\tan x}
| r = \ln \size x + \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n - 1} 2^{2 n} \paren {2^{2 n - 1} - 1} B_{2 n} \, x^{2 n} } {n \paren {2 n}!}
| c =
}}
{{eqn | r = \ln \size x + \frac {x^2} 3 + \frac {7 x^4} {90} + \frac {62 x^6} {2835} + \cdots
| c = ... | From [[Power Series Expansion for Cosecant Function]]:
{{begin-eqn}}
{{eqn | l = 2 \csc 2 x
| r = 2 \sum_{n \mathop = 0}^\infty \dfrac {\paren {-1}^{n - 1} 2 \paren {2^{2 n - 1} - 1} B_{2 n} \paren {2 x}^{2 n - 1} } {\paren {2 n}!}
| c =
}}
{{eqn | n = 1
| ll= \leadsto
| l = 2 \csc 2 x
| ... | Power Series Expansion for Logarithm of Tangent of x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_Tangent_of_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_Tangent_of_x | [
"Power Series Expansion for Logarithm of Tangent of x",
"Examples of Power Series",
"Logarithms",
"Tangent Function"
] | [] | [
"Power Series Expansion for Cosecant Function",
"Power Series is Termwise Integrable within Radius of Convergence",
"Definition:Primitive (Calculus)/Integration",
"Definition:Zeroth",
"Primitive of Cosecant of a x",
"Integral of Power",
"Primitive of Reciprocal"
] |
proofwiki-14285 | Power Series Expansion for Logarithm of 1 + x over 1 + x | {{begin-eqn}}
{{eqn | l = \frac {\map \ln {1 + x} } {1 + x}
| r = \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} H_n x^n
| c =
}}
{{eqn | r = x - H_2 x^2 + H_3 x^3 - H_4 x^4 + \cdots
| c =
}}
{{end-eqn}}
where $H_n$ denotes the $n$th harmonic number:
:$H_n = \ds \sum_{r \mathop = 1}^n \dfrac 1 r = ... | Let $\map f x = \dfrac {\map \ln {1 + x} } {1 + x}$.
By definition of Maclaurin series:
:$(1): \quad \map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$
where $\map {f^{\paren n} } 0$ denotes the $n$th derivative of $f$ {{WRT|Differentiation}} $x$ evaluated at $x = 0$.
From Nth Deriva... | {{begin-eqn}}
{{eqn | l = \frac {\map \ln {1 + x} } {1 + x}
| r = \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} H_n x^n
| c =
}}
{{eqn | r = x - H_2 x^2 + H_3 x^3 - H_4 x^4 + \cdots
| c =
}}
{{end-eqn}}
where $H_n$ denotes the $n$th [[Definition:Harmonic Number|harmonic number]]:
:$H_n = \ds \sum... | Let $\map f x = \dfrac {\map \ln {1 + x} } {1 + x}$.
By definition of [[Definition:Maclaurin Series|Maclaurin series]]:
:$(1): \quad \map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$
where $\map {f^{\paren n} } 0$ denotes the [[Definition:Higher Derivative|$n$th derivative]] of $... | Power Series Expansion for Logarithm of 1 + x over 1 + x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_1_+_x_over_1_+_x | https://proofwiki.org/wiki/Power_Series_Expansion_for_Logarithm_of_1_+_x_over_1_+_x | [
"Examples of Power Series",
"Logarithms"
] | [
"Definition:Harmonic Numbers"
] | [
"Definition:Maclaurin Series",
"Definition:Derivative/Higher Derivatives/Higher Order",
"Nth Derivative of Natural Logarithm by Reciprocal",
"Ratio Test",
"Definition:Absolutely Convergent Series"
] |
proofwiki-14286 | Derivative of Logarithm over Power | :$\dfrac \d {\d x} \dfrac {\ln x} {x^n} = \dfrac {1 - n \ln x} {x^{n + 1} }$ | {{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \dfrac {\ln x} {x^n}
| r = \dfrac \d {\d x} x^{-n} \ln x
| c =
}}
{{eqn | r = \ln x \dfrac \d {\d x} x^{-n} + x^{-n} \dfrac \d {\d x} \ln x
| c = Product Rule for Derivatives
}}
{{eqn | r = - n x^{-n - 1} \ln x + x^{-n} \dfrac 1 x
| c = Primitive of Re... | :$\dfrac \d {\d x} \dfrac {\ln x} {x^n} = \dfrac {1 - n \ln x} {x^{n + 1} }$ | {{begin-eqn}}
{{eqn | l = \dfrac \d {\d x} \dfrac {\ln x} {x^n}
| r = \dfrac \d {\d x} x^{-n} \ln x
| c =
}}
{{eqn | r = \ln x \dfrac \d {\d x} x^{-n} + x^{-n} \dfrac \d {\d x} \ln x
| c = [[Product Rule for Derivatives]]
}}
{{eqn | r = - n x^{-n - 1} \ln x + x^{-n} \dfrac 1 x
| c = [[Primitive... | Derivative of Logarithm over Power | https://proofwiki.org/wiki/Derivative_of_Logarithm_over_Power | https://proofwiki.org/wiki/Derivative_of_Logarithm_over_Power | [
"Derivatives involving Logarithm Functions",
"Logarithms"
] | [] | [
"Product Rule for Derivatives",
"Primitive of Reciprocal",
"Primitive of Power",
"Category:Derivatives involving Logarithm Functions",
"Category:Logarithms"
] |
proofwiki-14287 | Nth Derivative of Natural Logarithm by Reciprocal | :$\dfrac {\d^n} {\d x^n} \dfrac {\ln x} x = \paren {-1}^{n + 1} n! \dfrac {H_n - \ln x} {x^{n + 1} }$
where $H_n$ denotes the $n$th harmonic number:
:$H_n = \ds \sum_{r \mathop = 1}^n \dfrac 1 r = 1 + \dfrac 1 2 + \dfrac 1 3 + \cdots + \dfrac 1 r$ | The proof proceeds by induction.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$\dfrac {\d^n} {\d x^n} \dfrac {\ln x} x = \paren {-1}^{n + 1} n! \dfrac {H_n - \ln x} {x^{n + 1} }$ | :$\dfrac {\d^n} {\d x^n} \dfrac {\ln x} x = \paren {-1}^{n + 1} n! \dfrac {H_n - \ln x} {x^{n + 1} }$
where $H_n$ denotes the $n$th [[Definition:Harmonic Number|harmonic number]]:
:$H_n = \ds \sum_{r \mathop = 1}^n \dfrac 1 r = 1 + \dfrac 1 2 + \dfrac 1 3 + \cdots + \dfrac 1 r$ | The proof proceeds by [[Principle of Mathematical Induction|induction]].
For all $n \in \Z_{> 0}$, let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\dfrac {\d^n} {\d x^n} \dfrac {\ln x} x = \paren {-1}^{n + 1} n! \dfrac {H_n - \ln x} {x^{n + 1} }$ | Nth Derivative of Natural Logarithm by Reciprocal | https://proofwiki.org/wiki/Nth_Derivative_of_Natural_Logarithm_by_Reciprocal | https://proofwiki.org/wiki/Nth_Derivative_of_Natural_Logarithm_by_Reciprocal | [
"Derivatives involving Logarithm Functions",
"Natural Logarithms"
] | [
"Definition:Harmonic Numbers"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Principle of Mathematical Induction"
] |
proofwiki-14288 | Variance of Normal Distribution | Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
Then:
:$\var X = \sigma^2$ | From the definition of the normal distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
From Variance as Expectation of Square minus Square of Expectation:
:$\ds \var X = \int_{-\infty}^\infty x^2 \map {f_X} x \rd x... | Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the [[Definition:Normal Distribution|normal distribution]].
Then:
:$\var X = \sigma^2$ | From the definition of the [[Definition:Normal Distribution|normal distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
From [[Variance as Expectation of Square minus ... | Variance of Normal Distribution/Proof 1 | https://proofwiki.org/wiki/Variance_of_Normal_Distribution | https://proofwiki.org/wiki/Variance_of_Normal_Distribution/Proof_1 | [
"Variance of Normal Distribution",
"Variance",
"Normal Distribution"
] | [
"Definition:Normal Distribution"
] | [
"Definition:Normal Distribution",
"Definition:Probability Density Function",
"Variance as Expectation of Square minus Square of Expectation",
"Expectation of Normal Distribution",
"Integration by Substitution",
"Fundamental Theorem of Calculus",
"Gaussian Integral",
"Exponential Tends to Zero and Infi... |
proofwiki-14289 | Variance of Normal Distribution | Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
Then:
:$\var X = \sigma^2$ | By Moment Generating Function of Normal Distribution, the moment generating function of $X$ is given by:
:$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$
From Variance as Expectation of Square minus Square of Expectation:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
From Moment Generating Function of... | Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the [[Definition:Normal Distribution|normal distribution]].
Then:
:$\var X = \sigma^2$ | By [[Moment Generating Function of Normal Distribution]], the [[Definition:Moment Generating Function|moment generating function]] of $X$ is given by:
:$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$
From [[Variance as Expectation of Square minus Square of Expectation/Continuous|Variance as Expectation ... | Variance of Normal Distribution/Proof 2 | https://proofwiki.org/wiki/Variance_of_Normal_Distribution | https://proofwiki.org/wiki/Variance_of_Normal_Distribution/Proof_2 | [
"Variance of Normal Distribution",
"Variance",
"Normal Distribution"
] | [
"Definition:Normal Distribution"
] | [
"Moment Generating Function of Normal Distribution",
"Definition:Moment Generating Function",
"Variance as Expectation of Square minus Square of Expectation/Continuous",
"Moment Generating Function of Normal Distribution/Examples/Second Moment",
"Moment in terms of Moment Generating Function",
"Exponentia... |
proofwiki-14290 | Variance as Expectation of Square minus Square of Expectation/Continuous | Let $X$ be a continuous random variable.
Then the variance of $X$ can be expressed as:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$. | Let $\mu = \expect X$.
Let $X$ have probability density function $f_X$.
As $f_X$ is a probability density function:
:$\ds \int_{-\infty}^\infty \map {f_X} x \rd x = \Pr \paren {-\infty < X < \infty} = 1$
Then:
{{begin-eqn}}
{{eqn | l = \var X
| r = \expect {\paren {X - \mu}^2}
| c = {{Defof|Variance of Continuous ... | Let $X$ be a [[Definition:Continuous Random Variable|continuous random variable]].
Then the [[Definition:Variance of Continuous Random Variable|variance]] of $X$ can be expressed as:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
That is, it is the [[Definition:Expectation|expectation]] of the [[Definition:Squarin... | Let $\mu = \expect X$.
Let $X$ have [[Definition:Probability Density Function|probability density function]] $f_X$.
As $f_X$ is a [[Definition:Probability Density Function|probability density function]]:
:$\ds \int_{-\infty}^\infty \map {f_X} x \rd x = \Pr \paren {-\infty < X < \infty} = 1$
Then:
{{begin-eqn}... | Variance as Expectation of Square minus Square of Expectation/Continuous | https://proofwiki.org/wiki/Variance_as_Expectation_of_Square_minus_Square_of_Expectation/Continuous | https://proofwiki.org/wiki/Variance_as_Expectation_of_Square_minus_Square_of_Expectation/Continuous | [
"Variance as Expectation of Square minus Square of Expectation"
] | [
"Definition:Random Variable/Continuous",
"Definition:Variance/Continuous",
"Definition:Expectation",
"Definition:Square/Function",
"Definition:Square/Function",
"Definition:Expectation"
] | [
"Definition:Probability Density Function",
"Definition:Probability Density Function"
] |
proofwiki-14291 | Variance as Expectation of Square minus Square of Expectation/Discrete | Let $X$ be a discrete random variable.
Then the variance of $X$ can be expressed as:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
That is, it is the expectation of the square of $X$ minus the square of the expectation of $X$. | We let $\mu = \expect X$, and take the expression for variance:
:$\var X := \ds \sum_{x \mathop \in \Img X} \paren {x - \mu}^2 \map \Pr {X = x}$
Then:
{{begin-eqn}}
{{eqn | l = \var X
| r = \sum_x \paren {x^2 - 2 \mu x + \mu^2} \map \Pr {X = x}
| c =
}}
{{eqn | r = \sum_x x^2 \map \Pr {X = x} - 2 \mu \sum_... | Let $X$ be a [[Definition:Discrete Random Variable|discrete random variable]].
Then the [[Definition:Variance of Discrete Random Variable|variance]] of $X$ can be expressed as:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
That is, it is the [[Definition:Expectation|expectation]] of the [[Definition:Squaring|squa... | We let $\mu = \expect X$, and take the expression for [[Definition:Variance of Discrete Random Variable|variance]]:
:$\var X := \ds \sum_{x \mathop \in \Img X} \paren {x - \mu}^2 \map \Pr {X = x}$
Then:
{{begin-eqn}}
{{eqn | l = \var X
| r = \sum_x \paren {x^2 - 2 \mu x + \mu^2} \map \Pr {X = x}
| c =
}}
... | Variance as Expectation of Square minus Square of Expectation/Discrete | https://proofwiki.org/wiki/Variance_as_Expectation_of_Square_minus_Square_of_Expectation/Discrete | https://proofwiki.org/wiki/Variance_as_Expectation_of_Square_minus_Square_of_Expectation/Discrete | [
"Variance as Expectation of Square minus Square of Expectation"
] | [
"Definition:Random Variable/Discrete",
"Definition:Variance/Discrete",
"Definition:Expectation",
"Definition:Square/Function",
"Definition:Square/Function",
"Definition:Expectation"
] | [
"Definition:Variance/Discrete"
] |
proofwiki-14292 | Reversion of Power Series | Let $\ds y = \sum_{n \mathop = 1}^\infty c_n x^n$ be a power series.
Then:
:$\ds x = \sum_{n \mathop = 1}^\infty C_n y^n$
is also a power series, where:
{{begin-eqn}}
{{eqn | l = c_1 C_1
| r = 1
}}
{{eqn | l = {c_1}^3 C_2
| r = -c_2
}}
{{eqn | l = {c_1}^5 C_3
| r = 2 {c_2}^2 - c_1 c_3
}}
{{eqn | l = {... | {{ProofWanted}}
Plugging:
:$\ds x = \sum_{n \mathop = 1}^\infty C_n y^n$
into:
:$\ds y = \sum_{n \mathop = 1}^\infty c_n x^n$
the following equation is obtained
:$y = c_1 C_1 y + \paren {c_2 C_1^2 + c_1 C_2} y^2 + \paren {c_3 C_1^3 + 2 c_2 C_1 C_2 + c_1 C_3} y^3 + \paren {3 c_3 C_1^2 C_2 + c_2 C_2^2 + c_2 C_1 C_3} + \c... | Let $\ds y = \sum_{n \mathop = 1}^\infty c_n x^n$ be a [[Definition:Power Series|power series]].
Then:
:$\ds x = \sum_{n \mathop = 1}^\infty C_n y^n$
is also a [[Definition:Power Series|power series]], where:
{{begin-eqn}}
{{eqn | l = c_1 C_1
| r = 1
}}
{{eqn | l = {c_1}^3 C_2
| r = -c_2
}}
{{eqn | l = {... | {{ProofWanted}}
Plugging:
:$\ds x = \sum_{n \mathop = 1}^\infty C_n y^n$
into:
:$\ds y = \sum_{n \mathop = 1}^\infty c_n x^n$
the following equation is obtained
:$y = c_1 C_1 y + \paren {c_2 C_1^2 + c_1 C_2} y^2 + \paren {c_3 C_1^3 + 2 c_2 C_1 C_2 + c_1 C_3} y^3 + \paren {3 c_3 C_1^2 C_2 + c_2 C_2^2 + c_2 C_1 C_3} + \c... | Reversion of Power Series | https://proofwiki.org/wiki/Reversion_of_Power_Series | https://proofwiki.org/wiki/Reversion_of_Power_Series | [
"Power Series"
] | [
"Definition:Power Series",
"Definition:Power Series"
] | [] |
proofwiki-14293 | Expectation of Continuous Uniform Distribution | Let $a, b \in \R$ such that $a < b$.
Let $X \sim \ContinuousUniform a b$ be the continuous uniform distribution over $\closedint a b$.
Then:
:$\expect X = \dfrac {a + b} 2$ | From the definition of the continuous uniform distribution, $X$ has probability density function:
:$\map {f_X} x = \begin {cases} \dfrac 1 {b - a} & : a \le x \le b \\ 0 & : \text {otherwise} \end {cases}$
From the definition of the expected value of a continuous random variable:
:$\ds \expect X = \int_{-\infty}^\inf... | Let $a, b \in \R$ such that $a < b$.
Let $X \sim \ContinuousUniform a b$ be the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]] over $\closedint a b$.
Then:
:$\expect X = \dfrac {a + b} 2$ | From the definition of the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \begin {cases} \dfrac 1 {b - a} & : a \le x \le b \\ 0 & : \text {otherwise} \end {cases}$
From the definition ... | Expectation of Continuous Uniform Distribution | https://proofwiki.org/wiki/Expectation_of_Continuous_Uniform_Distribution | https://proofwiki.org/wiki/Expectation_of_Continuous_Uniform_Distribution | [
"Expectation",
"Continuous Uniform Distribution"
] | [
"Definition:Uniform Distribution/Continuous"
] | [
"Definition:Uniform Distribution/Continuous",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Primitive of Power",
"Fundamental Theorem of Calculus",
"Difference of Two Squares"
] |
proofwiki-14294 | Variance of Continuous Uniform Distribution | Let $a, b \in \R$ such that $a < b$.
Let $X \sim \ContinuousUniform a b$ be the continuous uniform distribution over $\closedint a b$.
Then:
:$\var X = \dfrac {\paren {b - a}^2} {12}$ | From the definition of the continuous uniform distribution, $X$ has probability density function:
:$\map {f_X} x = \begin{cases} \dfrac 1 {b - a} & a \le x \le b \\ 0 & \text{otherwise} \end{cases}$
From Variance as Expectation of Square minus Square of Expectation:
:$\ds \var X = \int_{-\infty}^\infty x^2 \map {f_X}... | Let $a, b \in \R$ such that $a < b$.
Let $X \sim \ContinuousUniform a b$ be the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]] over $\closedint a b$.
Then:
:$\var X = \dfrac {\paren {b - a}^2} {12}$ | From the definition of the [[Definition:Continuous Uniform Distribution|continuous uniform distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \begin{cases} \dfrac 1 {b - a} & a \le x \le b \\ 0 & \text{otherwise} \end{cases}$
From [[Variance as Expectat... | Variance of Continuous Uniform Distribution | https://proofwiki.org/wiki/Variance_of_Continuous_Uniform_Distribution | https://proofwiki.org/wiki/Variance_of_Continuous_Uniform_Distribution | [
"Variance",
"Continuous Uniform Distribution"
] | [
"Definition:Uniform Distribution/Continuous"
] | [
"Definition:Uniform Distribution/Continuous",
"Definition:Probability Density Function",
"Variance as Expectation of Square minus Square of Expectation/Continuous",
"Expectation of Continuous Uniform Distribution",
"Primitive of Power",
"Fundamental Theorem of Calculus",
"Difference of Two Powers/Exampl... |
proofwiki-14295 | Median of Normal Distribution | Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
Then the median of $X$ is equal to $\mu$. | From the definition of the normal distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
Note that $f_X$ is non-zero, sufficient to ensure a unique median.
By the definition of a median, to prove that $\mu$ is the med... | Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the [[Definition:Normal Distribution|normal distribution]].
Then the [[Definition:Median of Continuous Random Variable|median]] of $X$ is equal to $\mu$. | From the definition of the [[Definition:Normal Distribution|normal distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt {2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
Note that $f_X$ is non-zero, sufficient to ensu... | Median of Normal Distribution | https://proofwiki.org/wiki/Median_of_Normal_Distribution | https://proofwiki.org/wiki/Median_of_Normal_Distribution | [
"Normal Distribution",
"Medians"
] | [
"Definition:Normal Distribution",
"Definition:Median of Continuous Random Variable"
] | [
"Definition:Normal Distribution",
"Definition:Probability Density Function",
"Definition:Median of Continuous Random Variable",
"Definition:Median of Continuous Random Variable",
"Definition:Median of Continuous Random Variable",
"Integration by Substitution",
"Definite Integral of Even Function",
"Ga... |
proofwiki-14296 | Expectation of Gamma Distribution | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
The expectation of $X$ is given by:
:$\expect X = \dfrac \alpha \beta$ | From the definition of the Gamma distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\map \Gamma \alpha}$
From the definition of the expected value of a continuous random variable:
:$\ds \expect X = \int_0^\infty x \map {f_X} x \rd x$
So:
{{begin-eqn... | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the [[Definition:Gamma Distribution|Gamma distribution]].
The [[Definition:Expectation of Continuous Random Variable|expectation]] of $X$ is given by:
:$\expect X = \dfrac \alpha \beta$ | From the definition of the [[Definition:Gamma Distribution|Gamma distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {\beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\map \Gamma \alpha}$
From the definition of the [[Definition:Expectation of Continuous... | Expectation of Gamma Distribution/Proof 1 | https://proofwiki.org/wiki/Expectation_of_Gamma_Distribution | https://proofwiki.org/wiki/Expectation_of_Gamma_Distribution/Proof_1 | [
"Expectation of Gamma Distribution",
"Gamma Distribution",
"Expectation"
] | [
"Definition:Gamma Distribution",
"Definition:Expectation/Continuous"
] | [
"Definition:Gamma Distribution",
"Definition:Probability Density Function",
"Definition:Expectation/Continuous",
"Integration by Substitution",
"Gamma Difference Equation"
] |
proofwiki-14297 | Expectation of Gamma Distribution | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
The expectation of $X$ is given by:
:$\expect X = \dfrac \alpha \beta$ | By Moment Generating Function of Gamma Distribution, the moment generating function of $X$ is given by:
:$\map {M_X} t = \paren {1 - \dfrac t \beta}^{-\alpha}$
for $t < \beta$.
From Moment in terms of Moment Generating Function:
:$\expect X = \map { {M_X}'} 0$
From Moment Generating Function of Gamma Distribution: F... | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the [[Definition:Gamma Distribution|Gamma distribution]].
The [[Definition:Expectation of Continuous Random Variable|expectation]] of $X$ is given by:
:$\expect X = \dfrac \alpha \beta$ | By [[Moment Generating Function of Gamma Distribution]], the [[Definition:Moment Generating Function|moment generating function]] of $X$ is given by:
:$\map {M_X} t = \paren {1 - \dfrac t \beta}^{-\alpha}$
for $t < \beta$.
From [[Moment in terms of Moment Generating Function]]:
:$\expect X = \map { {M_X}'} 0$
F... | Expectation of Gamma Distribution/Proof 2 | https://proofwiki.org/wiki/Expectation_of_Gamma_Distribution | https://proofwiki.org/wiki/Expectation_of_Gamma_Distribution/Proof_2 | [
"Expectation of Gamma Distribution",
"Gamma Distribution",
"Expectation"
] | [
"Definition:Gamma Distribution",
"Definition:Expectation/Continuous"
] | [
"Moment Generating Function of Gamma Distribution",
"Definition:Moment Generating Function",
"Moment in terms of Moment Generating Function",
"Moment Generating Function of Gamma Distribution/Examples/First Moment"
] |
proofwiki-14298 | Variance of Gamma Distribution | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
The variance of $X$ is given by:
:$\var X = \dfrac \alpha {\beta^2}$ | From the definition of the Gamma distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\map \Gamma \alpha}$
From Variance as Expectation of Square minus Square of Expectation:
:$\ds \var X = \int_0^\infty x^2 \map {f_X} x \rd x - \paren {\expect X}^2$
... | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the [[Definition:Gamma Distribution|Gamma distribution]].
The [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \dfrac \alpha {\beta^2}$ | From the definition of the [[Definition:Gamma Distribution|Gamma distribution]], $X$ has [[Definition:Probability Density Function|probability density function]]:
:$\map {f_X} x = \dfrac {\beta^\alpha x^{\alpha - 1} e^{-\beta x} } {\map \Gamma \alpha}$
From [[Variance as Expectation of Square minus Square of Expectat... | Variance of Gamma Distribution/Proof 1 | https://proofwiki.org/wiki/Variance_of_Gamma_Distribution | https://proofwiki.org/wiki/Variance_of_Gamma_Distribution/Proof_1 | [
"Variance of Gamma Distribution",
"Variance",
"Gamma Distribution"
] | [
"Definition:Gamma Distribution",
"Definition:Variance"
] | [
"Definition:Gamma Distribution",
"Definition:Probability Density Function",
"Variance as Expectation of Square minus Square of Expectation/Continuous",
"Expectation of Gamma Distribution",
"Integration by Substitution",
"Gamma Difference Equation"
] |
proofwiki-14299 | Variance of Gamma Distribution | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the Gamma distribution.
The variance of $X$ is given by:
:$\var X = \dfrac \alpha {\beta^2}$ | By Moment Generating Function of Gamma Distribution, the moment generating function of $X$ is given by:
:$\map {M_X} t = \paren {1 - \dfrac t \beta}^{-\alpha}$
for $t < \beta$.
From Variance as Expectation of Square minus Square of Expectation:
:$\var X = \expect {X^2} - \paren {\expect X}^2$
From Expectation of Gam... | Let $X \sim \map \Gamma {\alpha, \beta}$ for some $\alpha, \beta > 0$, where $\Gamma$ is the [[Definition:Gamma Distribution|Gamma distribution]].
The [[Definition:Variance|variance]] of $X$ is given by:
:$\var X = \dfrac \alpha {\beta^2}$ | By [[Moment Generating Function of Gamma Distribution]], the [[Definition:Moment Generating Function|moment generating function]] of $X$ is given by:
:$\map {M_X} t = \paren {1 - \dfrac t \beta}^{-\alpha}$
for $t < \beta$.
From [[Variance as Expectation of Square minus Square of Expectation/Continuous|Variance as ... | Variance of Gamma Distribution/Proof 2 | https://proofwiki.org/wiki/Variance_of_Gamma_Distribution | https://proofwiki.org/wiki/Variance_of_Gamma_Distribution/Proof_2 | [
"Variance of Gamma Distribution",
"Variance",
"Gamma Distribution"
] | [
"Definition:Gamma Distribution",
"Definition:Variance"
] | [
"Moment Generating Function of Gamma Distribution",
"Definition:Moment Generating Function",
"Variance as Expectation of Square minus Square of Expectation/Continuous",
"Expectation of Gamma Distribution",
"Moment Generating Function of Gamma Distribution/Examples/Second Moment",
"Moment in terms of Momen... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.