Special function/Catalogs/Catalog

Special functions are mathematical functions that turn up so often that they have been named. This page lists the most common special functions by category, along with some of the properties that are important to functions belonging to each category. It must be stressed that there is no single way to categorize functions; any practical classification will contain overlapping categories.

Algebraic functions

Complex parts

Elementary transcendental functions

Name	Notation
Exponential function	$\exp(x)$ , $e^{x}$
Natural logarithm	$\log(x)$ , $\ln(x)$

Trigonometric functions

Name	Notation	Triangle formula	Exponential formula
Sine	$\sin(x)$	Opposite / Hypotenuse	$(e^{ix}-e^{-ix})/2i$
Cosine	$\cos(x)$	Adjacent / Hypotenuse	$(e^{ix}+e^{-ix})/2$
Tangent	$\tan(x)$	Opposite / Adjacent	$-{\mathit {i}}(e^{{\mathit {i}}x}-e^{-{\mathit {i}}x})/(e^{{\mathit {i}}x}+e^{-{\mathit {i}}x})$
Cosecant	$\csc(x)$	Hypotenuse / Opposite
Secant	$\sec(x)$	Hypotenuse / Adjacent
Cotangent	$\cot(x)$	Adjacent / Opposite

Hyperbolic functions

Name	Notation	Exponential formula
Hyperbolic sine	$\sinh(x)$	$(e^{x}-e^{-x})/2$
Hyperbolic cosine	$\cosh(x)$	$(e^{x}+e^{-x})/2$
Hyperbolic tangent	$\tanh(x)$	$(e^{x}-e^{-x})/(e^{x}+e^{-x})$
Hyperbolic cosecant	$\operatorname {csch} (x)$	$2/(e^{x}-e^{-x})$
Hyperbolic secant	$\operatorname {sech} (x)$	$2/(e^{x}+e^{-x})$
Hyperbolic cotangent	$\coth(x)$	$(e^{x}+e^{-x})/(e^{x}-e^{-x})$

Inverse trigonometric functions

Inverse hyperbolic functions

Name	Notation	Logarithmic formula
Inverse hyperbolic sine	$\operatorname {arcsinh} (x)$	$\ln {x+{\sqrt {x^{2}+1}}}$
Inverse hyperbolic cosine	$\operatorname {arccosh} (x)$	$\ln {x+{\sqrt {x^{2}-1}}}$
Inverse hyperbolic tangent	$\operatorname {arctanh} (x)$	${\frac {1}{2}}\ln {\frac {1+x}{1-x}}$
Inverse hyperbolic cosecant	$\operatorname {arccsch} (x)$
Inverse hyperbolic secant	$\operatorname {arcsech} (x)$
Inverse hyperbolic cotangent	$\operatorname {arccoth} (x)$

Other

Lambert W-function

Nonelementary integrals

Bessel function related

Elliptic integrals

Orthogonal polynomials

See catalog of orthogonal polynomials for a more detailed listing.

Name	Notation	Interval	Weight function	$f_{0}$ , $f_{1}$ , $f_{2}$ , $f_{3}$ , ...
Chebyshev (first kind)	$T_{n}$	$-1,1$	$(1-x^{2})^{-1/2}$	$1$ , $x$ , $2x^{2}-1$ , $4x^{3}-3x$ , ...
Chebyshev (second kind)	$U_{n}$	$-1,1$	$(1-x^{2})^{1/2}$	$1$ , $2x$ , $4x^{2}-1$ , $8x^{3}-4x$ , ...
Legendre	$P_{n}$	$-1,1$	$1$
Hermite	$H_{n}$	$-\infty ,\infty$	$e^{-x^{2}}$
Laguerre	$L_{n}$	$0,\infty$	$e^{-x}$
Associated Laguerre	$L_{n}^{(\alpha )}$	$0,\infty$	$x^{\alpha }e^{-x}$

Factorial and gamma related

Name	Notation	Discrete formula	Continuous formula
Factorial	$x!$	$1\cdot 2\cdot 3\cdots x$	$\Gamma (x+1)$
Gamma function	$\Gamma (x)$	$(x-1)!$	$\Gamma (x)$
Double factorial	$x!!$	$1\cdot 3\cdot 5\cdots x\;\;(x\;\mathrm {odd} )$ $2\cdot 4\cdot 6\cdots x\;\;(x\;\mathrm {even} )$	${\begin{matrix}{\frac {1}{\sqrt {\pi }}}\end{matrix}}\;2^{(x+1)/2}\;\Gamma (x/2+1)$
Binomial coefficient	$n \choose k$	${\frac {n!}{k!(n-k)!}}$	${\frac {\Gamma (n+1)}{\Gamma (k+1)\Gamma (n-k+1)}}$
Rising factorial	$(x)^{(n)}$	${\frac {(x+n-1)!}{(x-1)!}}$	${\frac {\Gamma (x+n)}{\Gamma (x)}}$
Falling factorial	$(x)_{(n)}$	${\frac {x!}{(x-n)!}}$	${\frac {\Gamma (x+1)}{\Gamma (x-n+1)}}$
Beta function	$\mathrm {\mathrm {B} } (x,y)$		${\frac {\Gamma (x)\Gamma (y)}{\Gamma (x+y)}}$
Harmonic number	$H_{n}$	$\textstyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}$	$\gamma +\psi (n+1)$
Digamma function	$\psi (x),\psi ^{(0)}(x)$	$H_{x-1}-\gamma$	${\begin{matrix}{\frac {d}{dx}}\end{matrix}}\ln \Gamma (x)$
Polygamma function (of order m)	$\psi ^{(m)}(x)$		$\left({\begin{matrix}{\frac {d}{dx}}\end{matrix}}\right)^{m+1}\ln \Gamma (x)$