hcju/proofwikips huggingface dataset:
Weierstrass's Elliptic Function is Even in z
Let $\omega_1$ and $\omega_2$ be non-zero complex constants with $\dfrac {\omega_1} {\omega_2}$ having a positive imaginary part.  For $z \in \C \setminus \left\{2m \omega_1 + 2n\omega_2: \left({n,m}\right) \in \Z^2\right\}$: :$\displaystyle \wp\left({-z; \omega_1, \omega_2}\right) = \wp\left({z; \omega_1, \omega_2}\right)$ That is, Weierstrass's elliptic function is even in $z$.
Duplication Formula for Weierstrass's Elliptic Function
Let $\omega_1$, $\omega_2$ be non-zero complex constants with $\dfrac {\omega_1} {\omega_2}$ having a positive imaginary part. Let $z$ be a complex number where $z \notin \set {2 m \omega_1 + 2 n \omega_2: \tuple {n, m} \in \Z^2}$. Then: :$\map \wp {2 z; \omega_1, \omega_2} = \dfrac 1 4 \paren {\dfrac {\map {\wp''} {z; \omega_1, \omega_2} } {\map {\wp'} {z; \omega_1, \omega_2} } }^2 - 2 \map \wp {z; \omega_1, \omega_2}$ where: :$\wp$ is Weierstrass's elliptic function  :$\wp'$ and $\wp''$ denote its first and second derivative with respect to $z$.
Integral to Infinity of Sine p x over x
:$\displaystyle \int_0^\infty \frac {\sin p x} x \rd x = \begin{cases} \dfrac \pi 2 & : p > 0 \\ 0 & : p = 0 \\ -\dfrac \pi 2 & : p < 0 \end{cases}$
Radical of Sum of Ideals
Let $A$ be a commutative ring with unity. Let $\mathfrak a, \mathfrak b \subseteq A$ be ideals. Then for the radical of their sum we have: :$\operatorname{Rad} \left({\mathfrak a + \mathfrak b}\right) = \operatorname{Rad} \left({\operatorname{Rad} \left({\mathfrak a}\right) + \operatorname{Rad} \left({\mathfrak b}\right)}\right)$
Ideal of Ring is Contained in Radical
Let $A$ be a commutative ring with unity. Let $\mathfrak a \subseteq A$ be an ideal. Then $\mathfrak a$ is contained in its radical: :$\mathfrak a \subseteq \operatorname{Rad} \left({\mathfrak a}\right)$
Radical of Ideal Preserves Inclusion
Let $A$ be a commutative ring with unity. Let $\mathfrak a \subseteq \mathfrak b \subseteq A$ be an ideals. Then we have an inclusion of their radicals: :$\operatorname{Rad} \left({\mathfrak a}\right) \subseteq \operatorname{Rad} \left({\mathfrak b}\right)$
Ideals with Coprime Radicals are Coprime
Let $A$ be a commutative ring with unity. Let $\mathfrak a, \mathfrak b \subseteq A$ be ideals. Let their radicals be coprime: :$\operatorname{Rad} \left({\mathfrak a}\right) + \operatorname{Rad} \left({\mathfrak b}\right) = \left({1}\right)$ Then $\mathfrak a$ and $\mathfrak b$ are coprime.
Radical of Unit Ideal
Let $A$ be a commutative ring with unity. Let $\left({1}\right)$ be its unit ideal. Then its radical equals $(1)$: :$\operatorname{Rad} \left({\left({1}\right)}\right) = \left({1}\right)$.
Radical of Power of Prime Ideal
Let $A$ be a commutative ring with unity. Let $\mathfrak p \subseteq A$ be a prime ideal. Let $n > 0$ be a natural number. Then the radical of the $n$th power of $\mathfrak p$ equals $\mathfrak p$: :$\operatorname{Rad} \left({\mathfrak p^n}\right) = \mathfrak p$.
Coefficients of Product of Two Polynomials
Let $R$ be a commutative ring with unity. Let $f, g \in R \sqbrk x$ be polynomials over $R$. For a natural number $n \ge 0$, let: :$a_n$ be the coefficient of the monomial $x^n$ in $f$. :$b_n$ be the coefficient of the monomial $x^n$ in $g$.
Equivalence of Definitions of Irreducible Polynomial over Field
Let $K$ be a field. {{TFAE|def = Irreducible Polynomial}}
Linear Combination of Integrals/Indefinite
:$\displaystyle \int \paren {\lambda \map f x + \mu \map g x} \rd x = \lambda \int \map f x \rd x + \mu \int \map g x \rd x$
Linear Combination of Integrals/Definite
:$\displaystyle \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$
Correspondence between Abelian Groups and Z-Modules/Isomorphism of Categories
Let $\Z$ be the ring of integers. Let $\mathbf{Ab}$ be the category of abelian groups. Let $\mathbf{\mathbb Z-Mod}$ be the category of unitary $\Z$-modules. Then the: * forgetful functor $\mathbf{\mathbb Z-Mod} \to \mathbf{Ab}$ * associated Z-module functor $\mathbf{Ab} \to \mathbf{\mathbb Z-Mod}$ are strict inverse functors. In particular, $\mathbf{Ab}$ and $\mathbf{\mathbb Z-Mod}$ are isomorphic.
Correspondence between Abelian Groups and Z-Modules/Homomorphisms
Let $G, H$ be abelian groups. Let $f : G \to H$ be a mapping.  # $f$ is a group homomorphism. # $f$ is a $\Z$-module homomorphism between the $\Z$-modules associated with $G$ and $H$.
Mapping to Singleton is Unique
Let $S$ be a set. Let $T$ be a singleton. Then there exists a unique mapping $S \to T$.
Rectangular Formula for Definite Integrals
Let $f$ be a real function which is integrable on the closed interval $\closedint a b$. Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$: :$\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$ Then the definite integral of $f$ {{WRT|Integration}} $x$ from $a$ to $b$ can be approximated as: :$\displaystyle \int_a^b \map f x \rd x \approx \sum_{r \mathop = 0}^{n - 1} h \map f {x_r}$ where $h = \dfrac {b - a} n$.
Trapezoidal Formula for Definite Integrals
Let $f$ be a real function which is integrable on the closed interval $\closedint a b$. Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$: :$\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$ Then the definite integral of $f$ {{WRT|Integration}} $x$ from $a$ to $b$ can be approximated as: :$\displaystyle \int_a^b \map f x \rd x \approx \dfrac h 2 \paren {\map f {x_0} + \map f {x_n} + \sum_{r \mathop = 1}^{n - 1} 2 \map f {x_r} }$ where $h = \dfrac {b - a} n$.
Simpson's Rule
Let $f$ be a real function which is integrable on the closed interval $\closedint a b$. Let $P = \set {x_0, x_1, x_2, \ldots, x_{n - 1}, x_n}$ form a normal subdivision of $\closedint a b$: :$\forall r \in \set {1, 2, \ldots, n}: x_r - x_{r - 1} = \dfrac {b - a} n$ where $n$ is even. Then the definite integral of $f$ {{WRT|Integration}} $x$ from $a$ to $b$ can be approximated as: :$\displaystyle \int_a^b \map f x \rd x \approx \dfrac h 3 \paren {\map f {x_0} + \map f {x_n} + \sum_{r \mathop = 1}^{m - 1} 2 \map f {x_{2 m - 1} } + \sum_{r \mathop = 1}^{m - 1} 4 \map f {x_{2 m} } }$ where: :$h = \dfrac {b - a} n$ :$m = \dfrac n 2$
Definite Integral to Infinity of Reciprocal of x Squared plus a Squared
:$\ds \int_0^\infty \dfrac {\d x} {x^2 + a^2} = \frac \pi {2 a}$ for $a \ne 0$.
Definite Integral to Infinity of Reciprocal of 1 plus Power of x/Corollary
:$\displaystyle \int_0^\infty \frac 1 {a^n + x^n} \rd x = \frac \pi {n a^{n - 1} } \csc \paren {\frac \pi n}$
Definite Integral to Infinity of Power of x over 1 + x
:$\displaystyle \int_0^\infty \dfrac {x^{p - 1} \rd x} {1 + x} = \frac \pi {\sin \pi p}$ for $0 < p < 1$.
Universal Property of Free Abelian Group on Set
Let $S$ be a set. Let $(\Z^{(S)}, \iota)$ be the free abelian group on $S$. Let $G$ be an abelian group. Let $f : S \to G$ be a mapping. Then there exists a unique group homomorphism $g : \Z^{(S)} \to G$ with $g \circ \iota = f$: :$\xymatrix{ S \ar[d]_\iota \ar[r]^{\forall f} & G\\ \Z^{(S)} \ar[ru]_{\exists ! g} }$
Definite Integral from 0 to a of Root of a Squared minus x Squared
:$\displaystyle \int_0^a \sqrt {a^2 - x^2} \rd x = \frac {\pi a^2} 4$ for $a > 0$.
Universal Property of Abelianization of Group
Let $G$ be a group. Let $G^{\operatorname {ab} }$ be its abelianization. Let $\pi : G \to G^{\operatorname {ab} }$ be the quotient group epimorphism. Let $H$ be an abelian group. Let $f: G \to H$ be a group homomorphism. Then there exists a unique group homomorphism $g : G^{\operatorname {ab}} \to H$ such that $g \circ \pi = f$: :$\xymatrix { G \ar[d]_\pi \ar[r]^{\forall f} & H\\ G^{\operatorname {ab} } \ar[ru]_{\exists ! g} }$
Definite Integral from 0 to a of x^m by (a^n - x^n)^p
:$\displaystyle \int_0^a x^m \paren {a^n - x^n}^p \rd x = \frac {a^{m + 1 + n p} \, \map \Gamma {\frac {m + 1} n} \map \Gamma {p + 1} } {n \map \Gamma {\frac {m + 1} n + p + 1} }$
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}}
Equivalent Characterizations of Finer Equivalence Relation
Let $X$ be a set. Let $\equiv$ and $\sim$ be equivalence relations on $X$.  # $\equiv$ 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 $\sim$-equivalence class is saturated under $\equiv$.
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}}
Definite Integral from 0 to Half Pi of Square of Sine x
:$\displaystyle \int_0^{\frac \pi 2} \sin^2 x \rd x = \frac \pi 4$
Definite Integral from 0 to Half Pi of Square of Cosine x
:$\displaystyle \int_0^{\frac \pi 2} \cos^2 x \rd x = \frac \pi 4$
Reduction Formula for Definite Integral of Power of Cosine
Let $n \in \Z_{> 0}$ be a positive integer. Let $I_n$ be defined as: :$\displaystyle I_n = \int_0^{\frac \pi 2} \cos^n x \rd x$ Then $\left\langle{I_n}\right\rangle$ is a decreasing sequence of real numbers which satisfies: :$n I_n = \left({n - 1}\right) I_{n - 2}$ Thus: :$I_n = \dfrac {n - 1} n  I_{n - 2}$ is a reduction formula for $I_n$.
Definite Integral from 0 to Half Pi of Even Power of Cosine x
:$\displaystyle \int_0^{\frac \pi 2} \cos^{2 n} x \rd x = \dfrac {\paren {2 n}!} {\paren {2^n n!}^2} \dfrac \pi 2$
Definite Integral from 0 to Half Pi of Odd Power of Cosine x
:$\displaystyle \int_0^{\frac \pi 2} \cos^{2 n + 1} x \rd x = \dfrac {\left({2^n n!}\right)^2} {\left({2 n + 1}\right)!}$
Arctangent of Zero is Zero
:$\arctan 0 = 0$
Riemann P-symbol in terms of Gaussian Hypergeometric Function
Let: :$\displaystyle f\left({z}\right) = \operatorname P \left\{ \begin{matrix} a & b & c \\ \alpha & \beta & \gamma & z \\ \alpha' & \beta' & \gamma' \end{matrix} \right\}$ where: :$\operatorname P$ is the Riemann P-symbol :$\alpha + \beta + \gamma + \alpha' + \beta' + \gamma' = 1$  :$\alpha - \alpha'$ is not a negative integer.  Then:  :$\displaystyle f\left({z}\right) = \left({ \frac {z - a} {z - b} }\right)^\alpha \left({ \frac {z - c} {z - b} }\right)^\gamma {}_2 \operatorname F_1 \left({ {\alpha + \beta + \gamma, \alpha + \beta' + \gamma} \atop {1 + \alpha - \alpha'} } \, \middle \vert {\, \frac {\left({z - a}\right) \left({c - b}\right)} {\left({z - b} \right) \left({c - a}\right)} }\right)$ where ${}_2 \operatorname F_1$ is the Gaussian hypergeometric function.
Sum of Powers of Positive Integers
Let $n, p \in \Z_{>0}$ be (strictly) positive integers. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^n k^p       | r = 1^p + 2^p + \cdots + n^p       | c =  }} {{eqn | r = \frac {n^{p + 1} } {p + 1} + \sum_{k \mathop = 1}^p \frac {B_k \, p^{\underline {k - 1} } \, n^{p - k + 1} } {k!}       | c =  }} {{eqn | r = \frac {n^{p + 1} } {p + 1} + \frac {B_1 \, n^p} {1!} + \frac {B_2 \, p \, n^{p - 1} } {2!} + \frac {B_4 \, p \paren {p - 1} \paren {p - 2} n^{p - 3} } {4!} + \cdots       | c =  }} {{end-eqn}} where: :$B_k$ are the Bernoulli numbers :$p^{\underline k}$ is the $k$th falling factorial of $p$.
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.
Sum of Reciprocals of Sixth Powers Alternating in Sign
{{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \dfrac {\paren {-1}^{n + 1} } {n^6}       | r = \frac 1 {1^6} - \frac 1 {2^6} + \frac 1 {3^6} - \frac 1 {4^6} + \cdots       | c =  }} {{eqn | r = \frac {31 \pi^6} {30 \, 240}       | c =  }} {{end-eqn}}
Sum of Reciprocals of Fourth Powers of Odd Integers
{{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {\left({2 n - 1}\right)^4}       | r = 1 + \dfrac 1 {3^4} + \dfrac 1 {5^4} + \dfrac 1 {7^4} + \dfrac 1 {9^4} + \cdots       | c =  }} {{eqn | r = \dfrac {\pi^4} {96}       | c =  }} {{end-eqn}}
Sum of Reciprocals of Sixth Powers of Odd Integers
{{begin-eqn}} {{eqn | l = \sum_{n \mathop = 1}^\infty \frac 1 {\left({2 n - 1}\right)^6}       | r = 1 + \dfrac 1 {3^6} + \dfrac 1 {5^6} + \dfrac 1 {7^6} + \dfrac 1 {9^6} + \cdots       | c =  }} {{eqn | r = \dfrac {\pi^6} {960}       | c =  }} {{end-eqn}}
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: :$\displaystyle \map f x \sim \frac 8 \pi \sum_{r \mathop = 0}^\infty \frac {\sin \paren {2 r + 1} x} {\paren {2 r + 1}^3}$
Sum of Reciprocals of Powers of Odd Integers Alternating in Sign
:$\displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s} = \frac 1 {2 \map \Gamma s} \int_0^\infty x^{s - 1} \map \sech x \rd x$ where: :$\map \Re s > 0$ :$\Gamma$ is the gamma function :$\sech$ is the hyperbolic secant function.
Sum of Reciprocals of Cubes of Odd Integers Alternating in Sign in Pairs
{{begin-eqn}} {{eqn | l = \frac 1 {1^3} + \frac 1 {3^3} - \frac 1 {5^3} - \frac 1 {7^3} + \cdots       | r = \frac {3 \pi^3 \sqrt 2} {128}       | c =  }} {{end-eqn}}
Sum of Sequence of Products of Consecutive Odd Reciprocals
{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^n \frac 1 {\paren {2 j + 1} \paren {2 j + 3} }       | r = \frac 1 {1 \times 3} + \frac 1 {3 \times 5} + \frac 1 {5 \times 7} + \frac 1 {7 \times 9} + \cdots + \frac 1 {\paren {2 n + 1} \paren {2 n + 3} }       | c =  }} {{eqn | r = \frac {n + 1} {2 n + 3}       | c =  }} {{end-eqn}}
Equivalence of Definitions of Local Ring Homomorphism
Let $(A, \mathfrak m)$ and $(B, \mathfrak n)$ be commutative local rings. Let $f : A \to B$ be a unital ring homomorphism. {{TFAE|def = Local Ring Homomorphism}}
Sum of Sequence of Products of Consecutive Odd and Consecutive Even Reciprocals
{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^n \frac 1 {j \left({j + 2}\right)}       | r = \frac 1 {1 \times 3} + \frac 1 {2 \times 4} + \frac 1 {3 \times 5} + \frac 1 {4 \times 6} + \cdots + \frac 1 {n \left({n + 2}\right)}       | c =  }} {{eqn | r = \frac 3 4 - \frac {2 n + 3} {2 \left({n + 1}\right) \left({n + 2}\right)}       | c =  }} {{end-eqn}}
Sum of Sequence of Products of Squares of 3 Consecutive Reciprocals
{{begin-eqn}} {{eqn | l = \sum_{j \mathop = 1}^\infty \frac 1 {j^2 \paren {j + 1}^2 \paren {j + 2}^2}       | r = \frac 1 {1^2 \times 2^2 \times 3^2} + \frac 1 {2^2 \times 3^2 \times 4^2} + \frac 1 {3^2 \times 4^2 \times 5^2} + \frac 1 {4^2 \times 5^2 \times 6^2} + \cdots       | c =  }} {{eqn | r = \frac {4 \pi^2 - 39} {16}       | c =  }} {{end-eqn}}
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 $a, d > 0$.
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 {2^{2 n} - 1} \pi^{2 n} } {2 \paren {2 n}!}       | c =  }} {{end-eqn}}
Sum of Reciprocals of 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 \left({-1}\right)^{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 = \left({-1}\right)^{n + 1} \dfrac {B_{2 n} \left({2^{2 n - 1} - 1}\right) \pi^{2 n} } {\left({2 n}\right)!}       | c =  }} {{end-eqn}}
Hermite's Formula for Hurwitz Zeta Function
:$\displaystyle \map \zeta {s, q} = \frac 1 {2 q^s} + \frac { q^{1 - s} } {s - 1} + 2 \int_0^\infty \frac {\map \sin {s \arctan \frac x q} } {\paren {q^2 + x^2}^{\frac 1 2 s} \paren {e^{2 \pi x} - 1} } \rd x$ where:  :$\zeta$ is the Hurwitz zeta function :$\map \Re s > 1$ :$\map \Re q > 0$.
Binet's Formula for Logarithm of Gamma Function/Formulation 1
Let $z$ be a complex number with a positive real part. Then:  :$\displaystyle \Ln \map \Gamma z = \paren {z - \frac 1 2} \Ln z - z + \frac 1 2 \ln 2 \pi + \int_0^\infty \paren {\frac 1 2 - \frac 1 t + \frac 1 {e^t - 1} } \frac {e^{-t z} } t \rd t$ where:  :$\Gamma$ is the Gamma function :$\Ln$ is the principal branch of the complex logarithm.
Binet's Formula for Logarithm of Gamma Function/Formulation 2
Let $z$ be a complex number with a positive real part. Then:  :$\displaystyle \Ln \map \Gamma z = \paren {z - \frac 1 2} \Ln z - z + \frac 1 2 \ln 2 \pi + 2 \int_0^\infty \frac {\map \arctan {t / z} } {e^{2 \pi t} - 1} \rd t$ where:  :$\Gamma$ is the Gamma function :$\Ln$ is the principal branch of the complex logarithm.
Sum of Reciprocals of Odd Powers of Odd Integers Alternating in Sign
Let $n \in \Z_{\ge 0}$ be a positive integer. Then: {{begin-eqn}} {{eqn | l = \sum_{j \mathop = 0}^\infty \frac {\left({-1}\right)^j} {\left({2 j + 1}\right)^{2 n + 1} }       | r = \left({-1}\right)^{n + 1} \frac {\pi^{2 n + 1} E_{2 n} } {2^{2 n + 2} \left({2 n}\right)!}       | c =  }} {{eqn | r = \frac 1 {1^{2 n + 1} } - \frac 1 {3^{2 n + 1} } + \frac 1 {5^{2 n + 1} } - \frac 1 {7^{2 n + 1} } + \cdots       | c =  }} {{end-eqn}} where $E_n$ is the $n$th Euler number.
Sum of Infinite Series of Product of Power and Cosine
Let $r \in \R$ such that $\size r < 1$. Let $z \in \R$ such that $z \ne 2 m \pi$ for any $m \in \Z$. 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}}
Sum of Series of Product of Power and Sine
Let $r \in \R$. Let $x \in \R$ such that $x \ne 2 m \pi$ for any $m \in \Z$. 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}       | c =  }} {{end-eqn}}
Euler-Maclaurin Summation Formula
Let $f$ be a real function which is appropriately differentiable and integrable. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 1}^{n - 1} \map f k       | r = \int_0^n \map f x \rd x - \frac {\map f 0 + \map f n} 2 + \sum_{k \mathop = 1}^\infty \frac {B_{2 k} } {\paren {2 k}!} \paren {\map {f^{\paren {2 k - 1} } } n - \map {f^{\paren {2 k - 1} } } 0}       | c =  }} {{eqn | r = \int_0^n \map f x \rd x - \frac 1 2 \paren {\map f n + \map f 0}       | c =  }} {{eqn | o =        | ro= +       | r = \frac 1 {12} \paren {\map {f'} n - \map {f'} 0}       | c =  }} {{eqn | o =        | ro= -       | r = \frac 1 {720} \paren {\map {f'''} n - \map {f'''} 0}       | c =  }} {{eqn | o =        | ro= +       | r = \frac 1 {30 \, 240} \paren {\map {f^{\paren 5} } n - \map {f^{\paren 5} } 0}       | c =  }} {{eqn | o =        | ro= -       | r = \frac 1 {1 \, 209 \, 600} \paren {\map {f^{\paren 7} } n - \map {f^{\paren 7} } 0}       | c =  }} {{end-eqn}} where: :$f^{\paren k}$ denotes the $k$th derivative of $f$ :$B_n$ denotes the $n$th Bernoulli number. {{Proofread|Confusion between the various types of Bernoulli numbers means I may have got signs and indices wrong.}}
Power Series Expansion of 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 \left({-1}\right)^k x^k       | c =  }} {{eqn | r = 1 - x + x^2 - x^3 + x^4 - \cdots       | c =  }} {{end-eqn}}
Frullani's Integral
:$\displaystyle \int_0^\infty \frac {\map f {a x} - \map f {b x} } x \rd x = \paren {\map f \infty - \map f 0} \ln \frac a b$
Power Series Expansion of 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}}
Power Series Expansion 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 = 1 + \sum_{k \mathop = 1}^\infty \left({-1}\right)^{k - 1} \frac {\left({2 \left({k - 1}\right)}\right)!} {2^{2 k - 1} k! \left({k - 1}\right)!} x^k       | c =  }} {{eqn | r = 1 + \frac 1 2 x - \frac 2 {2 \times 4} x^2 + \frac {1 \times 3} {2 \times 4 \times 6} x^3 - \cdots       | c =  }} {{end-eqn}}
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 $-1 < x < 1$.
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 all $x \in \R$ such that $-1 < x < 1$.
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 {\left({-1}\right)^{n} 2^{2 n} B_{2 n} \, x^{2 n - 1} } {\left({2 n}\right)!}       | c =  }} {{eqn | r = \frac 1 x - \frac x 3 - \frac {x^3} {45} - \frac {2 x^5} {945} + \cdots       | c =  }} {{end-eqn}} where $B_{2 n}$ denotes the Bernoulli numbers. This converges for $0 < \left|{x}\right| < \pi$.
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} + \cdots }} {{end-eqn}} where $E_{2 n}$ denotes the Euler numbers. This converges for $\size x < \dfrac \pi 2$.
Power Series Expansion for Real Arccotangent Function
The arccotangent function has a Taylor series expansion: :$\operatorname {arccot} x = \begin{cases} \displaystyle \frac \pi 2 - \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2 n + 1} } {2 n + 1} & : -1 \le x \le 1 \\ \displaystyle \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac 1 {\left({2 n + 1}\right) x^{2 n + 1} } & : x \ge 1 \\ \displaystyle \pi + \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac 1 {\left({2 n + 1}\right) x^{2 n + 1} } & : x \le -1 \end{cases}$ That is: :$\operatorname {arccot} x = \begin{cases} \displaystyle \frac \pi 2 - \left({x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \cdots}\right) & : -1 \le x \le 1 \\ \displaystyle \frac 1 x - \frac 1 {3 x^3} + \frac 1 {5 x^5} - \cdots & : x \ge 1 \\ \displaystyle \pi + \frac 1 x - \frac 1 {3 x^3} + \frac 1 {5 x^5} - \cdots & : x \le -1 \end{cases}$
Power Series Expansion for Real Arccosecant Function
The arccosecant function has a Taylor Series expansion: {{begin-eqn}} {{eqn | l = \operatorname {arccsc} x       | r = \sum_{n \mathop = 0}^\infty \frac {\left({2 n}\right)!} {2^{2 n} \left({n!}\right)^2 \left({2 n + 1}\right) x^{2 n + 1} }       | c =  }} {{eqn | r = \frac 1 x + \frac 1 2 \frac 1 {3 x^3} + \frac {1 \times 3} {2 \times 4} \frac 1 {5 x^5} + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} \frac 1 {7 x^7} + \cdots       | c =  }} {{end-eqn}} which converges for $\left\lvert{x}\right\rvert \ge 1$.