Special function/Catalogs/Catalog: Difference between revisions

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imported>Charles Blackham
(expansion of circular, hyperbolic & inv hyperbolic trig f'ns)
imported>Fredrik Johansson
(consistency; fix some tex markup)
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!Exponential formula
!Exponential formula
|-
|-
|[[Hyperbolic Sine]]
|[[Hyperbolic sine]]
|<math>\sinh(x)</math>
|<math>\sinh(x)</math>
|<math>(e^{x}-e^{-x})/2</math>
|<math>(e^{x}-e^{-x})/2</math>
|-
|-
|[[Hyperbolic Cosine]]
|[[Hyperbolic cosine]]
|<math>\cosh(x)</math>
|<math>\cosh(x)</math>
|<math>(e^{x}+e^{-x})/2</math>
|<math>(e^{x}+e^{-x})/2</math>
|-
|-
|[[Hyperbolic Tangent]]
|[[Hyperbolic tangent]]
|<math>\tanh(x)</math>
|<math>\tanh(x)</math>
|<math>(e^{x}-e^{-x})/(e^{x}+e^{-x})</math>
|<math>(e^{x}-e^{-x})/(e^{x}+e^{-x})</math>
|-
|-
|[[Hyperbolic Cosecant]]
|[[Hyperbolic cosecant]]
|<math>\operatorname{csch}(x)</math>
|<math>\operatorname{csch}(x)</math>
|<math>2/(e^{x}-e^{-x})</math>
|<math>2/(e^{x}-e^{-x})</math>
|-
|-
|[[Hyperbolic Secant]]
|[[Hyperbolic secant]]
|<math>\operatorname{sech}(x)</math>
|<math>\operatorname{sech}(x)</math>
|<math>2/(e^{x}+e^{-x})</math>
|<math>2/(e^{x}+e^{-x})</math>
|-
|-
|[[Hyperbolic Cotangent]]
|[[Hyperbolic cotangent]]
|<math>\coth(x)</math>
|<math>\coth(x)</math>
|<math>(e^{x}+e^{-x})/(e^{x}-e^{-x})</math>
|<math>(e^{x}+e^{-x})/(e^{x}-e^{-x})</math>
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!Logarithmic formula
!Logarithmic formula
|-
|-
|[[Inverse Hyperbolic Sine]]
|[[Inverse hyperbolic sine]]
|<math>\operatorname{arcsinh}(x)</math>
|<math>\operatorname{arcsinh}(x)</math>
|<math>\ln{x+\sqrt{x^2+1}}</math>
|<math>\ln{x+\sqrt{x^2+1}}</math>
|-
|-
|[[Inverse Hyperbolic Cosine]]
|[[Inverse hyperbolic cosine]]
|<math>\operatorname{arccosh}(x)</math>
|<math>\operatorname{arccosh}(x)</math>
|<math>\ln{x+\sqrt{x^2-1}}</math>
|<math>\ln{x+\sqrt{x^2-1}}</math>
|-
|-
|[[Inverse Hyperbolic Tangent]]
|[[Inverse hyperbolic tangent]]
|<math>\operatorname{arctanh}(x)</math>
|<math>\operatorname{arctanh}(x)</math>
|<math>\frac{1}{2}ln{\frac{1+x}{1-x}}</math>
|<math>\frac{1}{2}\ln{\frac{1+x}{1-x}}</math>
|-
|-
|[[Inverse Hyperbolic Cosecant]]
|[[Inverse hyperbolic cosecant]]
|<math>\operatorname{arccsch}(x)</math>
|<math>\operatorname{arccsch}(x)</math>
|
|
|-
|-
|[[Inverse Hyperbolic Secant]]
|[[Inverse hyperbolic secant]]
|<math>\operatorname{arcsech}(x)</math>
|<math>\operatorname{arcsech}(x)</math>
|
|
|-
|-
|[[Inverse Hyperbolic Cotangent]]
|[[Inverse hyperbolic cotangent]]
|<math>\operatorname{arccoth}(x)</math>
|<math>\operatorname{arccoth}(x)</math>
|
|
|}
|}


===Other===
===Other===
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|<math>\psi(x), \psi^{(0)}(x)</math>
|<math>\psi(x), \psi^{(0)}(x)</math>
|<math>H_{x-1}-\gamma</math>
|<math>H_{x-1}-\gamma</math>
|<math>\begin{matrix}\frac{d}{dx}\end{matrix} \log \Gamma(x)</math>
|<math>\begin{matrix}\frac{d}{dx}\end{matrix} \ln \Gamma(x)</math>
|-
|-
|[[Polygamma function]]<br/>(of order ''m'')
|[[Polygamma function]]<br/>(of order ''m'')
|<math>\psi^{(m)}(x)</math>
|<math>\psi^{(m)}(x)</math>
|
|
|<math>\left(\begin{matrix}\frac{d}{dx}\end{matrix}\right)^{m+1} \log \Gamma(x)</math>
|<math>\left(\begin{matrix}\frac{d}{dx}\end{matrix}\right)^{m+1} \ln \Gamma(x)</math>
|}
|}


Revision as of 14:54, 25 April 2007

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 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)} ,
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 \operatorname{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 \operatorname{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

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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \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

Nonelementary integrals

Bessel function related

Elliptic 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} , , , ...
Legendre
Hermite
Laguerre
Associated Laguerre

Factorial and gamma related

Name Notation Discrete formula Continuous formula
Factorial
Gamma function
Double factorial

Binomial coefficient
Rising factorial
Falling factorial
Beta function
Harmonic number
Digamma function
Polygamma function
(of order m)

Notes:

Zeta function related

Hypergeometric functions

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

See also

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