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

Complex parts

Elementary transcendental functions

Name Notation
Exponential function ,
Natural logarithm ,

Trigonometric functions:

Name Notation Triangle formula Exponential formula
Sine Opposite / Hypotenuse
Cosine Adjacent / Hypotenuse
Tangent Opposite / Adjacent
Cosecant Hypotenuse / Opposite
Secant Hypotenuse / Adjacent
Cotangent Adjacent / Opposite

Hyperbolic functions:

Name Notation Exponential formula
Hyperbolic sine
Hyperbolic cosine
Hyperbolic tangent
Hyperbolic cosecant
Hyperbolic secant
Hyperbolic cotangent

Inverse trigonometric functions:

Name Notation Triangle formula Exponential formula
Arcsine
Arccosine
Arctangent
Arccosecant
Arcsecant
Arccotangent


Inverse hyperbolic functions:

Name Notation Logarithmic formula
Inverse hyperbolic sine
Inverse hyperbolic cosine
Inverse hyperbolic tangent
Inverse hyperbolic cosecant
Inverse hyperbolic secant
Inverse hyperbolic cotangent

Other:

Exponential integral related

Function Notation Definition
Exponential integral
Logarithmic integral

Trigonometric integrals:

Function Notation Definition
Sine integral
Hyperbolic sine integral
Cosine integral
Hyperbolic cosine integral

Note: is Euler's constant

Related to the normal distribution:

Name Notation Definition
Gaussian function none standardized
Error function
Complementary error function

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 , , , , ...
Chebyshev (first kind) , , , , ...
Chebyshev (second kind) , , , , ...
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: