# Special function/Catalogs/Catalog

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.

## Elementary transcendental functions

Name Notation
Exponential function , 
Natural logarithm , 
Name Notation Triangle formula Exponential formula
Sine  Opposite / Hypotenuse 
Cosine  Adjacent / Hypotenuse 
Tangent  Opposite / Adjacent 
Cosecant  Hypotenuse / Opposite
Secant  Hypotenuse / Adjacent
Cotangent  Adjacent / Opposite
Name Notation Exponential formula
Hyperbolic sine  
Hyperbolic cosine  
Hyperbolic tangent  
Hyperbolic cosecant  
Hyperbolic secant  
Hyperbolic cotangent  
Name Notation Triangle formula Exponential formula
Arcsine 
Arccosine 
Arctangent 
Arccosecant 
Arcsecant 
Arccotangent 
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  
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.

## 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:

## Hypergeometric functions

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