Special function/Catalogs/Catalog: Difference between revisions

Latest revision as of 13:58, 8 December 2009

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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

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)} , 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:

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

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)	$U_{n}$	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} , 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^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 8x^3 - 4x} , ...
Legendre	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 P_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}	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 {\textstyle \frac{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 (3x^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 {\textstyle \frac{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 (5x^3-3x)} , …
Hermite	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 H_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 -\infty,\infty}	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^2}}
Laguerre	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 L_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 0,\infty}	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}}
Associated Laguerre	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 L_n^{(\alpha)}}	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 0,\infty}	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^{\alpha} e^{-x}}

Factorial and gamma related

Name	Notation	Discrete formula	Continuous formula
Factorial	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 1 \cdot 2 \cdot 3 \cdots 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 \Gamma(x+1)}
Gamma 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 \Gamma(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 (x-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 \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)	$\psi ^{(m)}(x)$		$\left({\begin{matrix}{\frac {d}{dx}}\end{matrix}}\right)^{m+1}\ln \Gamma (x)$

Notes:

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

Zeta function related

Hypergeometric functions

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

@@ Line 1: / Line 1: @@
+{{subpages}}
 [[Special function]]s are mathematical [[function (mathematics)|function]]s 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.
@@ Line 27: / Line 28: @@
 |<math>\exp(x)</math>, <math>e^x</math>
 |-
-|[[Natural logarithm]]
+|[[Logarithm|Natural logarithm]]
 |<math>\log(x)</math>, <math>\ln(x)</math>
 |}
-===Trigonometric functions===
+[[Trigonometric function]]s:
-{| class="wikitable"
+{| class="wikitable" style="margin-top:0"
 !Name
 !Notation
@@ Line 69: / Line 70: @@
 |}
-===Hyperbolic functions===
+[[Hyperbolic function]]s:
-{| class="wikitable"
+{| class="wikitable" style="margin-top:0"
 !Name
 !Notation
@@ Line 88: / Line 89: @@
 |-
 |[[Hyperbolic cosecant]]
-|<math>\operatorname{csch}(x)</math>
+|<math>\mathrm{csch}(x)</math>
 |<math>2/(e^{x}-e^{-x})</math>
 |-
 |[[Hyperbolic secant]]
-|<math>\operatorname{sech}(x)</math>
+|<math>\mathrm{sech}(x)</math>
 |<math>2/(e^{x}+e^{-x})</math>
 |-
@@ Line 100: / Line 101: @@
 |}
-===Inverse trigonometric functions===
+[[Inverse trigonometric function]]s:
+{| class="wikitable" style="margin-top:0"
+!Name
+!Notation
+!Triangle formula
+!Exponential formula
+|-
+|[[Arcsine]]
+|<math>\arcsin(x)</math>
+|
+|
+|-
+|[[Arccosine]]
+|<math>\arccos(x)</math>
+|
+|
+|-
+|[[Arctangent]]
+|<math>\arctan(x)</math>
+|
+|
+|-
+|[[Arccosecant]]
+|<math>\arccsc(x)</math>
+|
+|
+|-
+|[[Arcsecant]]
+|<math>\arcsec(x)</math>
+|
+|
+|-
+|[[Arccotangent]]
+|<math>\arccot(x)</math>
+|
+|
+|}
-===Inverse hyperbolic functions===
+[[Inverse hyperbolic function]]s:
-{| class="wikitable"
+{| class="wikitable" style="margin-top:0"
 !Name
 !Notation
@@ Line 109: / Line 147: @@
 |-
 |[[Inverse hyperbolic sine]]
-|<math>\operatorname{arcsinh}(x)</math>
+|<math>\mathrm{arcsinh}(x)</math>
-|<math>\ln{x+\sqrt{x^2+1}}</math>
+|<math>\ln{(x+\sqrt{x^2+1)}}</math>
 |-
 |[[Inverse hyperbolic cosine]]
-|<math>\operatorname{arccosh}(x)</math>
+|<math>\mathrm{arccosh}(x)</math>
-|<math>\ln{x+\sqrt{x^2-1}}</math>
+|<math>\ln{(x+\sqrt{x^2-1})}</math>
 |-
 |[[Inverse hyperbolic tangent]]
-|<math>\operatorname{arctanh}(x)</math>
+|<math>\mathrm{arctanh}(x)</math>
 |<math>\frac{1}{2}\ln{\frac{1+x}{1-x}}</math>
 |-
 |[[Inverse hyperbolic cosecant]]
-|<math>\operatorname{arccsch}(x)</math>
+|<math>\mathrm{arccsch}(x)</math>
 |
 |-
 |[[Inverse hyperbolic secant]]
-|<math>\operatorname{arcsech}(x)</math>
+|<math>\mathrm{arcsech}(x)</math>
 |
 |-
 |[[Inverse hyperbolic cotangent]]
-|<math>\operatorname{arccoth}(x)</math>
+|<math>\mathrm{arccoth}(x)</math>
 |
 |}
-===Other===
+Other:
+* [[Sinc function]]
 * [[Lambert W-function]]
@@ Line 233: / Line 272: @@
 |<math>-1,1</math>
 |<math>1</math>
-|
+|<math>1</math>, <math>x</math>, <math>{\textstyle \frac{1}{2}}</math><math>(3x^2-1)</math>, <math>{\textstyle \frac{1}{2}}</math><math>(5x^3-3x)</math>, &hellip;
 |-
 |[[Hermite polynomials|Hermite]]
@@ Line 275: / Line 314: @@
 |<math>1 \cdot 3 \cdot 5 \cdots x \;\;(x \; \mathrm{odd})</math><br/>
 <math>2 \cdot 4 \cdot 6 \cdots x \;\;(x \; \mathrm{even})</math>
-|
+|<math>\frac{\Gamma(x+1)}{2^\frac{x-1}2 *\Gamma(\frac{x+1}2)}\;\;(x \; \mathrm{odd})</math>
+<br/><math>2^\frac{x-1}2 * \Gamma(\frac{x+1}2) \;\;(x \; \mathrm{even}) </math>
 |-
 |[[Binomial coefficient]]
@@ Line 332: / Line 372: @@
 * [[Hypergeometric function]]s
 * [[Meijer G-function]]
-==See also==
-* [[Catalog of mathematical constants]]
-* [[Catalog of probability distributions]]
-* [[Catalog of number sequences]]
-==Further reading==
-* Introductory material: {{cite book | author = N. N. Lebedev | title = Special Functions and their applications | publisher = Dover | date = 1972 | address = New York}}
-==References==
-* {{cite book | author = Milton Abramowitz and Irene A. Stegun | title = Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables | publisher = Dover | date = 1964 | address = New York}} ([http://www.math.sfu.ca/~cbm/aands/ available online])
-* {{cite book | author = I. S. Gradstein and I. M. Ryzhik | title = Table of integrals, series and products | publisher = Academic Press | date = 2000 | address = London}}
-* {{cite book | author = A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi | title = Higher Transcendental Functions (Vol I and II) | publisher = McGraw-Hill Book Company | date = 1953 | address = New York - Toronto - London}}
-[[Category:CZ Live]]
-[[Category:Mathematics Workgroup]]

Special function/Catalogs/Catalog: Difference between revisions

Latest revision as of 13:58, 8 December 2009

A - For a New Cluster use the following directions

B - For a Cluster Move use the following directions

Algebraic functions

Complex parts

Elementary transcendental functions

Exponential integral related

Bessel function related

Elliptic integrals

Orthogonal polynomials

Factorial and gamma related

Zeta function related

Hypergeometric functions

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