# 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

In mathematics, most functions are defined as a solution of a differential equation.[1] 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.[1][5]

## Examples

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

## References

1. Andrews, G. E., Askey, R., & Roy, R. (1999). Special functions (Vol. 71). Cambridge University Press.
2. Silverman, R. A. (1972). Special functions and their applications. Courier Corporation.
3. Nikiforov, A. F., & Uvarov, V. B. (1988). Special functions of mathematical physics (Vol. 205). Basel: Birkhäuser.
4. Gil, A., Segura, J., & Temme, N. M. (2007). Numerical methods for special functions. Society for Industrial and Applied Mathematics.
5. 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.
6. Davis, P. J. (1959). Leonhard euler's integral: A historical profile of the gamma function. The American Mathematical Monthly, 66(10), 849-869.
7. Artin, E. (2015). The gamma function. Courier Dover Publications.
8. Gautschi, W. (2004). Orthogonal polynomials. Oxford: Oxford University Press.
9. Cohl, H. S., & Ismail, M. E. (Eds.). (2020). Lectures on Orthogonal Polynomials and Special Functions (Vol. 464). Cambridge University Press.
10. Ismail, M., Ismail, M. E., & van Assche, W. (2005). Classical and quantum orthogonal polynomials in one variable (Vol. 13). Cambridge University Press.
11. Higham, N. J. (2008). Functions of matrices: theory and computation. Society for Industrial and Applied Mathematics.