Special functions
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.[1][2][3] Most of them appear in higher education. Some experts are studying numerical methods for them.[4]
Definition change
In mathematics, most functions are defined as a solution of a differential equation.[1] 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.[1][5]
Examples change
- Gamma function, it is studied since Euler[1][6][7]
- Orthogonal polynomials, these are polynomials with special properties.[8][9][10]
- Matrix functions, these are studied in linear algebra and matrix analysis.[11]
For more examples, find textbooks named "special functions".
References change
- ↑ 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.
Other websites change
- 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