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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
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:
Inverse trigonometric functions:
Inverse hyperbolic functions:
Other:
Exponential integral related
Trigonometric integrals:
Note: is Euler's constant
Related to the normal distribution:
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
Notes:
Zeta function related
Hypergeometric functions
Note: many of the preceding functions are special cases of the following: