Definition[change | change source]
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.
Examples[change | change source]
- 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".
References[change | change source]
- 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.
Other websites[change | change source]
- 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