Special function/Catalogs/Catalog

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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	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 \exp(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 e^x}
Natural logarithm	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 \log(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 \ln(x)}

Trigonometric functions:

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

Hyperbolic functions:

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

Inverse trigonometric functions:

Name	Notation	Triangle formula	Exponential formula
Arcsine	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 \arcsin(x)}
Arccosine	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 \arccos(x)}
Arctangent	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 \arctan(x)}
Arccosecant	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 \arccsc(x)}
Arcsecant	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 \arcsec(x)}
Arccotangent	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 \arccot(x)}

Inverse hyperbolic functions:

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

Other:

• Sinc function
• Lambert W-function

Exponential integral related

Function	Notation	Definition
Exponential integral	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 \mathrm{Ei}(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 \textstyle -\int_{-x}^{\infty} \frac{e^{-t}}{t} \, dt}
Logarithmic integral	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 \mathrm{li}(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 \textstyle \int_0^x \frac{1}{\ln t} \, dt}

Trigonometric integrals:

Function	Notation	Definition
Sine integral	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 \mathrm{Si}(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 \textstyle \int_0^x \frac{\sin t}{t} \, dt}
Hyperbolic sine integral	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 \mathrm{Shi}(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 \textstyle \int_0^x \frac{\sinh t}{t} \, dt}
Cosine integral	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 \mathrm{Ci}(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 \textstyle \gamma + \ln x + \int_0^x \frac{\cos t - 1}{t} \, dt}
Hyperbolic cosine integral	$\mathrm {Chi} (x)$	$\textstyle \gamma +\ln x+\int _{0}^{x}{\frac {\cosh t-1}{t}}\,dt$

Note: $\gamma$ is Euler's constant

Related to the normal distribution:

Name	Notation	Definition
Gaussian function	none standardized	$f(x)=ae^{-(x-b)^{2}/c^{2}}$
Error function	$\mathrm {erf} (x)$	$\textstyle {\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}dt$
Complementary error function	$\mathrm {erfc} (x)$	$1-\mathrm {erf} (x)$

See also gamma related functions below; in particular, the incomplete gamma functions.

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	$f_{0}$, $f_{1}$, $f_{2}$, $f_{3}$, ...
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	$P_{n}$	$-1,1$	$1$	$1$, $x$, ${\textstyle {\frac {1}{2}}}$$(3x^{2}-1)$, ${\textstyle {\frac {1}{2}}}$$(5x^{3}-3x)$, …
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	$x!!$	$1\cdot 3\cdot 5\cdots x\;\;(x\;\mathrm {odd} )$ $2\cdot 4\cdot 6\cdots x\;\;(x\;\mathrm {even} )$	${\frac {\Gamma (x+1)}{2^{\frac {x-1}{2}}*\Gamma ({\frac {x+1}{2}})}}\;\;(x\;\mathrm {odd} )$ $2^{\frac {x-1}{2}}*\Gamma ({\frac {x+1}{2}})\;\;(x\;\mathrm {even} )$
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)	$\psi ^{(m)}(x)$		$\left({\begin{matrix}{\frac {d}{dx}}\end{matrix}}\right)^{m+1}\ln \Gamma (x)$

• 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