Special functions

Special functions are some mathematical functions used in mathematical analysis or physics. Most of them appear in higher education. Some experts are studying numerical methods for them.

Definition

In mathematics, most functions are defined as a solution of a differential equation. For example, the exponential function ${\displaystyle \exp(x)}$ is the solution of the ordinary differential equation ${\displaystyle y^{\prime }=y}$. Due to this relation, some mathematicians are studying the connection between ODEs and special functions.

Examples

• Gamma function, it is studied since Euler
• Orthogonal polynomials, these are polynomials with special properties.
• Matrix functions, these are studied in linear algebra and matrix analysis.
• Riemann zeta function, related to the distribution of prime numbers in the Riemann hypothesis.
• Gauss error function, a function often used in probability theory

For more examples, find textbooks named "special functions".

References

Other websites

• National Institute of Standards and Technology, United States Department of Commerce. NIST Digital Library of Mathematical Functions. Archived from the original on December 13, 2018.
• Eric W. Weisstein, Special Function at MathWorld.
• Special functions at EqWorld: The World of Mathematical Equations
• Special functions and polynomials by Gerard 't Hooft and Stefan Nobbenhuis (April 8, 2013)
• Numerical Methods for Special Functions, by A. Gil, J. Segura, N.M. Temme (2007).
• R. Jagannathan, (P,Q)-Special Functions
• Specialfunctionswiki Archived 2025-01-12 at the Wayback Machine, a wiki about special functions