The term ``special function'' is not a completely precise mathematical term. It usually refers to a function of one or more real or complex variables which is either of use in some application or interesting in its own right, and hence has been studied enough to warrant giving it a name. Special functions are usually named after the mathematician who first introduced them or contributed much to their theory although, as in the rest of mathematics, such attributions are not always accurate, and they should be taken with a grain of salt.

Elementary Functions

• exponential
• logarithm
• trigonometric functions
• cyclometric functions
• hyperbolic functions
• area functions
• $\operatorname{sinc}$ function

Antiderivatives of elementary functions

• error function
• logarithmic integral
• exponential integral
• cosine integral
• sine integral
• hyperbolic sine integral
• Fresnel integrals
• elliptic integrals

Gamma and related functions

• gamma function
• beta function
• polygamma functions
• Barnes function

Functions defined as solutions of linear differential equations

• Bessel functions
• hypergeometric function
• confluent hypergeometric function
• generalized hypergeometric functions
• Hermite polynomials
• Legendre polynomials
• associated Laguerre polynomials
• spherical harmonics
• surface harmonics
• Lamé function
• Heun function

Functions defined as solutions of non-linear equations

• Painlevé transcendents
• Emden function
• Lambert W function

Abelian functions

• Jacobi theta functions
• Weierstrass sigma function
• Jacobi Zeta function
• Weierstrass zeta function

Zeta Functions

• general Zeta functions (in the sense of Jorgensen and Lang)
• Hecke zeta function
• Hurwitz zeta function
• Hecke Zeta function
• $L$ -functions
• Riemann zeta function
• Zeta functions of surfaces
• Zeta functions of graphs
• Zeta functions of operators