functions which have established names and notations due to their importance in mathematics
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.
In mathematics, most functions are defined as a solution of a differential equation. For example, the exponential function is the solution of the ordinary differential equation . Due to this relation, some mathematicians are studying the connection between ODEs and special functions.
- 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.
For more examples, find textbooks named "special functions".
- ↑ 1.0 1.1 1.2 1.3 Andrews, G. E., Askey, R., & Roy, R. (1999). Special functions (Vol. 71). Cambridge University Press.
- ↑ Silverman, R. A. (1972). Special functions and their applications. Courier Corporation.
- ↑ Nikiforov, A. F., & Uvarov, V. B. (1988). Special functions of mathematical physics (Vol. 205). Basel: Birkhäuser.
- ↑ Gil, A., Segura, J., & Temme, N. M. (2007). Numerical methods for special functions. Society for Industrial and Applied Mathematics.
- ↑ Iwasaki, K., Kimura, H., Shimemura, S., & Yoshida, M. (2013). From Gauss to Painlevé: a modern theory of special functions (Vol. 16). Springer Science & Business Media.
- ↑ Davis, P. J. (1959). Leonhard euler's integral: A historical profile of the gamma function. The American Mathematical Monthly, 66(10), 849-869.
- ↑ Artin, E. (2015). The gamma function. Courier Dover Publications.
- ↑ Gautschi, W. (2004). Orthogonal polynomials. Oxford: Oxford University Press.
- ↑ Cohl, H. S., & Ismail, M. E. (Eds.). (2020). Lectures on Orthogonal Polynomials and Special Functions (Vol. 464). Cambridge University Press.
- ↑ Ismail, M., Ismail, M. E., & van Assche, W. (2005). Classical and quantum orthogonal polynomials in one variable (Vol. 13). Cambridge University Press.
- ↑ Higham, N. J. (2008). Functions of matrices: theory and computation. Society for Industrial and Applied Mathematics.
- 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, a wiki about special functions