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

• Polynomials
  • Linear functions
  • Quadratic functions
  • Cubic functions
• Rational functions
• Power functions
  • Square root
  • Cube root

Complex parts

• Absolute value
• Real part
• Imaginary part
• Complex conjugate
• Complex argument
• Sign function

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 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	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{arccoth}(x)}

Other

• Lambert W-function

Nonelementary integrals

• Exponential integral
• Logarithmic integral
• Trigonometric integrals
  • Sine integral
  • Cosine integral
  • Hyperbolic sine integral
  • Hyperbolic cosine integral

Bessel function related

• Airy functions
• Bessel functions
• Fresnel integrals

Orthogonal polynomials

See catalog of orthogonal polynomials for a more detailed listing.

Name	Notation	Interval	Weight function	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_0} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_3} , ...
Chebyshev (first kind)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1,1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2)^{-1/2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x^2 - 1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4x^3 - 3x} , ...
Chebyshev (second kind)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1,1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-x^2)^{1/2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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	$\psi (x),\psi ^{(0)}(x)$	$H_{x-1}-\gamma$	${\begin{matrix}{\frac {d}{dx}}\end{matrix}}\log \Gamma (x)$
Polygamma function (of order m)	$\psi ^{(m)}(x)$		$\left({\begin{matrix}{\frac {d}{dx}}\end{matrix}}\right)^{m+1}\log \Gamma (x)$

• Incomplete gamma function
• Incomplete beta function
• Regularized gamma function
• Regularized beta function
• Barnes G-function

Notes:

• $\gamma$ is Euler's constant
• The polygamma functions are generalized to continuous m by the Hurwitz zeta function

Zeta function related

• Riemann zeta function
• Hurwitz zeta function
• Polylogarithm

Hypergeometric functions

Note: many of the preceding functions are special cases of the following:

• Hypergeometric functions
• Meijer G-function

See also

• Catalog of mathematical constants
• Catalog of probability distributions
• Catalog of number sequences