hcju/proofwikips huggingface dataset:
Definition:Inverse Sine/Complex/Definition 2
Let $z \in \C$ be a complex number. The '''inverse sine of $z$''' is the multifunction defined as: :$\sin^{-1} \paren z := \set {\dfrac 1 i \ln \paren {i z + \sqrt {\cmod {1 - z^2} } \exp \paren {\dfrac i 2 \arg \paren {1 - z^2} } } + 2 k \pi: k \in \Z}$ where: : $\sqrt {\cmod {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$ : $\arg \paren {1 - z^2}$ denotes the argument of $1 - z^2$ : $\ln$ is the complex natural logarithm considered as a  multifunction.
Definition:Inverse Sine/Complex/Arcsine
The principal branch of the complex inverse sine function is defined as: :$\map \arcsin z = \dfrac 1 i \, \map \Ln {i z + \sqrt {1 - z^2} }$ where: : $\Ln$ denotes the principal branch of the complex natural logarithm : $\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.
Definition:Inverse Cosine/Real
Let $x \in \R$ be a real number such that $-1 \le x \le 1$. The '''inverse cosine of $x$''' is the multifunction defined as: :$\cos^{-1} \left({x}\right) := \left\{{y \in \R: \cos \left({y}\right) = x}\right\}$ where $\cos \left({y}\right)$ is the cosine of $y$.
Definition:Inverse Cosine/Complex
{{:Definition:Inverse Cosine/Complex/Definition 1}}
Definition:Inverse Cosine/Complex/Definition 1
Let $z \in \C$ be a complex number. The '''inverse cosine of $z$''' is the multifunction defined as: :$\cos^{-1} \left({z}\right) := \left\{{w \in \C: \cos \left({w}\right) = z}\right\}$ where $\cos \left({w}\right)$ is the cosine of $w$.
Definition:Inverse Cosine/Complex/Definition 2
Let $z \in \C$ be a complex number. The '''inverse cosine of $z$''' is the multifunction defined as: :$\cos^{-1} \left({z}\right) := \left\{{\dfrac 1 i \ln \left({z + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)} }\right) + 2 k \pi: k \in \Z}\right\}$ where: : $\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$ : $\arg \left({z^2 - 1}\right)$ denotes the argument of $z^2 - 1$ : $\ln$ denotes the complex natural logarithm considered as a  multifunction.
Definition:Inverse Cosine/Complex/Arccosine
The principal branch of the complex inverse cosine function is defined as: :$\map \arccos z = \dfrac 1 i \, \map \Ln {z + \sqrt {z^2 - 1} }$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.
Definition:Inverse Tangent/Real
Let $x \in \R$ be a real number. The '''inverse tangent of $x$''' is the multifunction defined as: :$\tan^{-1} \left({x}\right) := \left\{{y \in \R: \tan \left({y}\right) = x}\right\}$ where $\tan \left({y}\right)$ is the tangent of $y$.
Definition:Inverse Tangent/Complex
{{:Definition:Inverse Tangent/Complex/Definition 1}}
Definition:Inverse Tangent/Complex/Definition 1
The '''inverse tangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \tan^{-1} \left({z}\right) := \left\{{w \in \C: \tan \left({w}\right) = z}\right\}$ where $\tan \left({w}\right)$ is the tangent of $w$.
Definition:Inverse Tangent/Complex/Definition 2
The '''inverse tangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \tan^{-1} \paren z := \set {\dfrac 1 {2 i} \ln \paren {\dfrac {i - z} {i + z} } + k \pi: k \in \Z}$ where $\ln$ denotes the complex natural logarithm as a multifunction.
Definition:Inverse Tangent/Complex/Arctangent
The principal branch of the complex inverse tangent function is defined as: :$\map \arctan z := \dfrac 1 {2 i} \, \map \Ln {\dfrac {i - z} {i + z} }$ where $\Ln$ denotes the principal branch of the complex natural logarithm.
Definition:Inverse Cotangent/Complex
{{:Definition:Inverse Cotangent/Complex/Definition 1}}
Definition:Inverse Cotangent/Complex/Definition 1
The '''inverse cotangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \cot^{-1} \left({z}\right) := \left\{{w \in \C: \cot \left({w}\right) = z}\right\}$ where $\cot \left({w}\right)$ is the cotangent of $w$.
Definition:Inverse Cotangent/Complex/Definition 2
The '''inverse cotangent''' is a multifunction defined on $S$ as: :$\forall z \in S: \cot^{-1} \left({z}\right) := \left\{{\dfrac 1 {2 i} \ln \left({\dfrac {z + i} {z - i}}\right) + k \pi: k \in \Z}\right\}$ where $\ln$ denotes the complex natural logarithm as a multifunction.
Definition:Inverse Cotangent/Complex/Arccotangent
The principal branch of the complex inverse cotangent function is defined as: :$\map \arccot z := \dfrac 1 {2 i} \, \map \Ln {\dfrac {z + i} {z - i} }$ where $\Ln$ denotes the principal branch of the complex natural logarithm.
Definition:Inverse Secant/Real
Let $x \in \R$ be a real number such that $x \le -1$ or $x \ge 1$. The '''inverse secant of $x$''' is the multifunction defined as: :$\map {\sec^{-1} } x := \set {y \in \R: \map \sec y = x}$ where $\map \sec y$ is the secant of $y$.
Definition:Inverse Secant/Complex
{{:Definition:Inverse Secant/Complex/Definition 1}}
Definition:Inverse Secant/Complex/Definition 1
Let $z \in \C_{\ne 0}$ be a non-zero complex number. The '''inverse secant of $z$''' is the multifunction defined as: :$\map {\sec^{-1} } z := \set {w \in \C: \map \sec w = z}$ where $\map \sec w$ is the secant of $w$.
Definition:Inverse Secant/Complex/Definition 2
Let $z \in \C_{\ne 0}$ be a non-zero complex number. The '''inverse secant of $z$''' is the multifunction defined as: :$\sec^{-1} z := \set {\dfrac 1 i \map \ln {\dfrac {1 + \sqrt {\size {1 - z^2} } e^{\paren {i / 2} \map \arg {1 - z^2} } } z} + 2 k \pi: k \in \Z}$ where: :$\sqrt {\size {1 - z^2} }$ denotes the positive square root of the complex modulus of $1 - z^2$ :$\map \arg {1 - z^2}$ denotes the argument of $1 - z^2$ :$\ln$ denotes the complex natural logarithm as a multifunction.
Definition:Inverse Secant/Complex/Arcsecant
The principal branch of the complex inverse secant function is defined as: :$\forall z \in \C_{\ne 0}: \map \arcsec z := \dfrac 1 i \, \map \Ln {\dfrac {1 + \sqrt {1 - z^2} } z}$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.
Definition:Inverse Cosecant/Complex
{{:Definition:Inverse Cosecant/Complex/Definition 1}}
Definition:Inverse Cosecant/Complex/Definition 1
Let $z \in \C_{\ne 0}$ be a non-zero complex number. The '''inverse cosecant of $z$''' is the multifunction defined as: :$\csc^{-1} \left({z}\right) := \left\{{w \in \C: \csc \left({w}\right) = z}\right\}$ where $\csc \left({w}\right)$ is the cosecant of $w$.
Definition:Inverse Cosecant/Complex/Definition 2
Let $z \in \C_{\ne 0}$ be a non-zero complex number. The '''inverse cosecant of $z$''' is the multifunction defined as: :$\csc^{-1} \left({z}\right) := \left\{{\dfrac 1 i \ln \left({\dfrac {i + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)}} z}\right) + 2 k \pi: k \in \Z}\right\}$ where: : $\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$ : $\arg \left({z^2 - 1}\right)$ denotes the argument of $z^2 - 1$ : $\ln$ denotes the complex natural logarithm considered as a  multifunction.
Definition:Inverse Cosecant/Complex/Arccosecant
The principal branch of the complex inverse cosecant function is defined as: :$\forall z \in \C_{\ne 0}: \map \arccsc z := \dfrac 1 i \, \map \Ln {\dfrac {i + \sqrt {z^2 - 1} } z}$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.
Definition:Inverse Hyperbolic Sine/Complex/Arcsine
The principal branch of the complex inverse hyperbolic sine function is defined as: :$\forall z \in \C: \map {\Sinh^{-1} } z := \map \Ln {z + \sqrt {z^2 + 1} }$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$.
Definition:Inverse Hyperbolic Sine/Arcsine
{{:Definition:Complex Hyperbolic Arcsine}}
Definition:Inverse Hyperbolic Sine/Real/Definition 2
The '''inverse hyperbolic sine''' $\sinh^{-1}: \R \to \R$ is a real function defined on $\R$ as: :$\forall x \in \R: \map {\sinh^{-1} } x := \map \ln {x + \sqrt {x^2 + 1} }$ where: :$\ln$ denotes the natural logarithm of a (strictly positive) real number :$\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$.
Definition:Inverse Hyperbolic Cosine/Real/Definition 1
The '''inverse hyperbolic cosine''' $\cosh^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \cosh^{-1} \left({x}\right) := y \in \R_{\ge 0}: x = \cosh \left({y}\right)$ where $\cosh \left({y}\right)$ denotes the hyperbolic cosine function.
Definition:Inverse Hyperbolic Cotangent/Real/Definition 1
The '''inverse hyperbolic cotangent''' $\coth^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \coth^{-1} x := y \in \R: x = \coth y$ where $\coth y$ denotes the hyperbolic cotangent function.
Definition:Inverse Hyperbolic Secant/Real/Definition 1
The '''inverse hyperbolic secant''' $\operatorname{sech}^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \operatorname{sech}^{-1} \left({x}\right) := y \in \R_{\ge 0}: x = \operatorname{sech} \left({y}\right)$ where $\operatorname{sech} \left({y}\right)$ denotes the hyperbolic secant function.
Definition:Inverse Hyperbolic Cosecant/Real/Definition 1
The '''inverse hyperbolic cosecant''' $\csch^{-1}: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as: :$\forall x \in \R_{\ne 0}: \map {\csch^{-1} } x := y \in \R: x = \map \csch y$ where $\map \csch y$ denotes the hyperbolic cosecant function of $y$.
Definition:Inverse Hyperbolic Cosine/Complex/Arccosine
The principal branch of the complex inverse hyperbolic cosine function is defined as: :$\forall z \in \C: \map {\Cosh^{-1} } z := \map \Ln {z + \sqrt {z^2 - 1} }$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.
Definition:Inverse Hyperbolic Cosecant/Complex/Arccosecant
The principal branch of the complex inverse hyperbolic cosecant function is defined as: :$\forall z \in \C_{\ne 0}: \map {\Csch^{-1} } z := \map \Ln {\dfrac {1 + \sqrt {z^2 + 1} } z}$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {z^2 + 1}$ denotes the principal square root of $z^2 + 1$.
Definition:Inverse Hyperbolic Secant/Complex/Arcsecant
The principal branch of the complex inverse hyperbolic secant function is defined as: :$\forall z \in \C: \map {\Sech^{-1} } z := \map \Ln {\dfrac {1 + \sqrt{1 - z^2} } z}$ where: :$\Ln$ denotes the principal branch of the complex natural logarithm :$\sqrt {1 - z^2}$ denotes the principal square root of $1 - z^2$.
Definition:Inverse Hyperbolic Tangent/Complex/Arctangent
The principal branch of the complex inverse hyperbolic tangent function is defined as: :$\forall z \in \C: \map {\Tanh^{-1} } z := \dfrac 1 2 \, \map \Ln {\dfrac {1 + z} {1 - z} }$ where $\Ln$ denotes the principal branch of the complex natural logarithm.
Definition:Inverse Hyperbolic Cotangent/Complex/Arccotangent
The principal branch of the complex inverse hyperbolic cotangent function is defined as: :$\forall z \in \C: \map {\Coth^{-1} } z := \dfrac 1 2 \, \map \Ln {\dfrac {z + 1} {z - 1} }$ where $\Ln$ denotes the principal branch of the complex natural logarithm.
Definition:Inverse Hyperbolic Cosine/Real/Definition 2
The '''inverse hyperbolic cosine''' $\cosh^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \map {\cosh^{-1} } x := \map \ln {x + \sqrt {x^2 - 1} }$ where: :$\sqrt {x^2 - 1}$ denotes the positive square root of $x^2 - 1$ :$\ln$ denotes the natural logarithm of a (strictly positive) real number.
Definition:Inverse Hyperbolic Tangent/Real/Definition 1
The '''inverse hyperbolic tangent''' $\tanh^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \tanh^{-1} \left({x}\right) := y \in \R: x = \tanh \left({y}\right)$ where $\tanh \left({y}\right)$ denotes the hyperbolic tangent function.
Definition:Inverse Hyperbolic Sine/Real/Definition 1
The '''inverse hyperbolic sine''' $\sinh^{-1}: \R \to \R$ is a real function defined on $\R$ as: :$\forall x \in \R: \map {\sinh^{-1} } x := y \in \R: x = \map \sinh y$ where $\map \sinh y$ denotes the hyperbolic sine function.
Definition:Inverse Hyperbolic Cosine/Real
=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosine/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosine/Real/Definition 2}}
Definition:Image (Set Theory)/Mapping/Mapping/Definition 1
The '''image''' of a mapping $f: S \to T$ is the set: :$\Img f = \set {t \in T: \exists s \in S: \map f s = t}$ That is, it is the set of values taken by $f$.
Definition:Image (Set Theory)/Mapping/Mapping/Definition 2
The '''image''' of a mapping $f: S \to T$ is the set: :$\Img f = f \sqbrk S$ where $f \sqbrk S$ is the image of $S$ under $f$.
Definition:Inverse Hyperbolic Tangent/Real
Let $S$ denote the open real interval: : $S := \left({-1 \,.\,.\, 1}\right)$ === Definition 1 === {{:Definition:Inverse Hyperbolic Tangent/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Tangent/Real/Definition 2}}
Definition:Inverse Hyperbolic Cotangent/Real
=== Definition 1 === {{:Definition:Inverse Hyperbolic Cotangent/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cotangent/Real/Definition 2}}
Definition:Inverse Hyperbolic Secant/Real
=== Definition 1 === {{:Definition:Inverse Hyperbolic Secant/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Secant/Real/Definition 2}}
Definition:Inverse Hyperbolic Cosecant/Real
=== Definition 1 === {{:Definition:Inverse Hyperbolic Cosecant/Real/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Cosecant/Real/Definition 2}}
Definition:Dominate (Set Theory)/Definition 1
Then '''$S$ is dominated by $T$'''  there exists an injection from $S$ to $T$.
Definition:Dominate (Set Theory)/Definition 2
Then '''$S$ is dominated by $T$'''  $S$ is equivalent to some subset of $T$. That is,  there exists a bijection $f: S \to T'$ for some $T' \subseteq T$.
Definition:Dominate (Set Theory)/Strictly Dominate
'''$S$ is strictly dominated by set $T$'''  $S \preccurlyeq T$ but $\neg T \preccurlyeq S$. This can be written $S \prec T$ or $S < T$.
Definition:Exponentiation of Cardinals
Let $A$ and $B$ be sets. Let $\mathbf a$ and $\mathbf b$ be the cardinals associated respectively with $A$ and $B$. Then the '''exponentiation''' of $\mathbf a$ to $\mathbf b$ is defined as: :$\mathbf a^{\mathbf b} := \operatorname{Card} \left({A^B}\right)$ where: :$A^B$ denotes the set of all mappings from $B$ to $A$ :$\operatorname{Card} \left({A^B}\right)$ denotes the cardinal associated with $A^B$.
Definition:Inverse Hyperbolic Tangent/Real/Definition 2
The '''inverse hyperbolic tangent''' $\tanh^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \map {\tanh^{-1} } x := \dfrac 1 2 \map \ln {\dfrac {1 + x} {1 - x} }$ where $\ln$ denotes the natural logarithm of a (strictly positive) real number.
Definition:Inverse Hyperbolic Cotangent/Real/Definition 2
The '''inverse hyperbolic cotangent''' $\coth^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \coth^{-1} x := \dfrac 1 2 \map \ln {\dfrac {x + 1} {x - 1} }$ where $\ln$ denotes the natural logarithm of a (strictly positive) real number.
Definition:Inverse Hyperbolic Secant/Real/Definition 2
The '''inverse hyperbolic secant''' $\sech^{-1}: S \to \R$ is a real function defined on $S$ as: :$\forall x \in S: \map {\sech^{-1} } x := \map \ln {\dfrac {1 + \sqrt {1 - x^2} } x}$ where: :$\sqrt {1 - x^2}$ denotes the positive square root of $1 - x^2$ :$\ln$ denotes the natural logarithm of a (strictly positive) real number.
Definition:Inverse Hyperbolic Cosecant/Real/Definition 2
The '''inverse hyperbolic cosecant''' $\csch^{-1}: \R_{\ne 0} \to \R$ is a real function defined on the non-zero real numbers $\R_{\ne 0}$ as: :$\forall x \in \R_{\ne 0}: \map {\csch^{-1} } x := \map \ln {\dfrac 1 x + \dfrac {\sqrt {x^2 + 1} } {\size x} }$ where: :$\sqrt {x^2 + 1}$ denotes the positive square root of $x^2 + 1$ :$\ln$ denotes the natural logarithm of a (strictly positive) real number.
Definition:Real Number/Axiomatic Definition
Let $\struct {R, +, \times, \le}$ be a Dedekind complete ordered field. Then $R$ is called the '''(field of) real numbers'''. ==== Real Number Axioms ==== {{:Definition:Real Number/Axioms}}
Definition:Natural Numbers/Inductive Sets in Real Numbers
Let $\R$ be the set of real numbers. Let $\II$ be the set of all inductive sets defined as subsets of $\R$. Then the '''natural numbers''' $\N$ are defined as: :$\N := \displaystyle \bigcap \II$ where $\displaystyle \bigcap$ denotes intersection.
Definition:Square Root/Positive Real
Let $x \in \R_{\ge 0}$ be a positive real number. The '''square roots of $x$''' are the real numbers defined as: :$x^{\paren {1 / 2} } := \set {y \in \R: y^2 = x}$ where $x^{\paren {1 / 2} }$ is the $2$nd root of $x$. The notation: :$y = \pm \sqrt x$ is usually encountered.
Definition:Naturally Ordered Semigroup
The concept of a '''naturally ordered semigroup''' is intended to capture the behaviour of the natural numbers $\N$, addition $+$ and the ordering $\le$ as they pertain to $\N$. === Naturally Ordered Semigroup Axioms === {{:Definition:Naturally Ordered Semigroup/Axioms}}
Definition:Square Root/Positive
The '''positive square root of $x$''' is the number defined as: :$+ \sqrt x := y \in \R_{>0}: y^2 = x$
Definition:Square Root/Negative
The '''negative square root of $x$''' is the number defined as: :$- \sqrt x := y \in \R_{<0}: y^2 = x$
Definition:Square Root/Negative Real
Let $x \in \R_{< 0}$ be a (strictly) negative real number. Then the '''square root of $x$''' is defined as: :$\sqrt x = i \paren {\pm \sqrt {-x} }$ where $i$ is the imaginary unit: :$i^2 = -1$
Definition:Square Root/Complex Number
{{:Definition:Square Root/Complex Number/Definition 1}}
Definition:Square Root/Complex Number/Definition 1
Let $z \in \C$ be a complex number expressed in polar form as $\left \langle{r, \theta}\right\rangle = r \left({\cos \theta + i \sin \theta}\right)$. The '''square root of $z$''' is the $2$-valued multifunction: {{begin-eqn}} {{eqn | l = z^{1/2}       | r = \left\{ {\sqrt r \left({\cos \left({\frac {\theta + 2 k \pi} 2}\right) + i \sin \left({\frac {\theta + 2 k \pi} 2}\right) }\right): k \in \left\{ {0, 1}\right\} }\right\}       | c =  }} {{eqn | r = \left\{ {\sqrt r \left({\cos \left({\frac \theta 2 + k \pi}\right) + i \sin \left({\frac \theta 2 + k \pi}\right) }\right): k \in \left\{ {0, 1}\right\} }\right\}       | c =  }} {{end-eqn}} where $\sqrt r$ denotes the positive square root of $r$.
Definition:Square Root/Complex Number/Definition 2
Let $z \in \C$ be a complex number expressed in polar form as $\left \langle{r, \theta}\right\rangle = r \left({\cos \theta + i \sin \theta}\right)$. The '''square root of $z$''' is the $2$-valued multifunction: :$z^{1/2} = \left\{ {\pm \sqrt r \left({\cos \left({\dfrac \theta 2}\right) + i \sin \left({\dfrac \theta 2}\right) }\right)}\right\}$ where $\pm \sqrt r$ denotes the positive and negative square roots of $r$.
Definition:Square Root/Complex Number/Definition 3
Let $z \in \C$ be a complex number. The '''square root of $z$''' is the $2$-valued multifunction: :$z^{1/2} = \left\{ {\sqrt {\left\vert{z}\right\vert} e^{\left({i / 2}\right) \arg \left({z}\right)} }\right\}$ where: : $\sqrt {\left\vert{z}\right\vert}$ denotes the positive square root of the complex modulus of $z$ : $\arg \left({z}\right)$ denotes the argument of $z$ considered as a  multifunction.
Definition:Square Root/Complex Number/Definition 4
Let $z \in \C$ be a complex number. The '''square root of $z$''' is the $2$-valued multifunction: :$z^{1/2} = \left\{ {w \in \C: w^2 = z}\right\}$
Definition:Gamma Function/Integral Form
The '''Gamma function''' $\Gamma: \C \to \C$ is defined, for the open right half-plane, as: :$\displaystyle \map \Gamma z = \map {\MM \set {e^{-t} } } z = \int_0^{\to \infty} t^{z - 1} e^{-t} \rd t$ where $\MM$ is the Mellin transform. For all other values of $z$ except the non-positive integers, $\map \Gamma z$ is defined as: :$\map \Gamma {z + 1} = z \, \map \Gamma z$
Definition:Gamma Function/Weierstrass Form
The '''{{AuthorRef|Karl Theodor Wilhelm Weierstrass|Weierstrass}} form''' of the '''Gamma function''' is: :$\displaystyle \frac 1 {\map \Gamma z} = z e^{\gamma z} \prod_{n \mathop = 1}^\infty \paren {\paren {1 + \frac z n} e^{-z / n} }$ where $\gamma$ is the Euler-Mascheroni constant. The '''Weierstrass form''' is valid for all $\C$.
Definition:Gamma Function/Euler Form
The '''{{AuthorRef|Leonhard Paul Euler|Euler}} form''' of the '''Gamma function''' is: :$\displaystyle \Gamma \left({z}\right) = \frac 1 z \prod_{n \mathop = 1}^\infty \left({\left({1 + \frac 1 n}\right)^z \left({1 + \frac z n}\right)^{-1}}\right) = \lim_{m \mathop \to \infty} \frac {m^z m!} {z \left({z + 1}\right) \left({z + 2}\right) \cdots \left({z + m}\right)}$ which is valid except for $z \in \left\{{0, -1, -2, \ldots}\right\}$.
Definition:Square Root/Complex Number/Principal Square Root
Let $z \in \C$ be a complex number. Let $z^{1/2} = \set {w \in \C: w^2 = z}$ be the square root of $z$. The '''principal square root''' of $z$ is the element $w$ of $z^{1/2}$ such that: :$\begin{cases} \map \Im w > 0 : & \map \Im z \ne 0 \\ \map \Re w \ge 0 : & \map \Im z = 0 \end{cases}$
Definition:Reduction Formula
Let $f: \R \times \Z_{\ge 0}: \R$ be a Darboux integrable real-valued function on $\R \times \N$. Let: :$\displaystyle I_n = \int \map f {x, n} \rd x$ for some $n \in \Z_{\ge 0}$. A '''reduction formula''' is a recurrence relation of the form: :$\displaystyle I_k = \map g {\displaystyle I_n}$ such that $k < n$. That is, it is a technique to '''reduce''' the integer parameter in the integrand in order to allow evaluation of the integral at, usually, $n = 0$ or $n = 1$. This technique relies upon the supposition that $\map f {x, 0}$ or $\map f {x, 1}$ can be integrated directly.
Definition:Zero Complement
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup. Let $0$ be the zero of $S$. Let $S^* := \relcomp S {\set 0} = S \setminus \set 0$ be the complement of $\set 0$ in $S$. Then $S^*$ is called the '''zero complement''' of $S$.
Definition:Beta Function
{{:Definition:Beta Function/Definition 1}}
Definition:Subtraction/Naturally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Let $m, n \in S$ such that $m \preceq n$. By axiom $(NO3)$, there exists a $p \in S$ such that: :$m + p = n$ This $p$ is the '''difference between $m$ and $n$''', and denoted $n - m$. The operation $-$, assigning to $m, n \in S$ with $m \preceq n$ their '''difference''' $n - m$ is called '''subtraction'''.
Definition:Bernoulli Numbers
The '''Bernoulli numbers''' $B_n$ are a sequence of rational numbers defined by:  === Generating Function === {{:Definition:Bernoulli Numbers/Generating Function}} === Recurrence Relation === {{:Definition:Bernoulli Numbers/Recurrence Relation}}
Definition:Convex Real Function/Definition 1
$f$ is '''convex on $I$''' : :$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) \le \alpha f \left({x}\right) + \beta f \left({y}\right)$
Definition:Convex Real Function/Definition 2
$f$ is '''convex on $I$''' : :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \le \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$
Definition:Convex Real Function/Definition 3
$f$ is '''convex on $I$''' : :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \le \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$
Definition:Concave Real Function/Definition 1
$f$ is '''concave on $I$''' : :$\forall x, y \in I: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) \ge \alpha f \left({x}\right) + \beta f \left({y}\right)$