Special functions

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

• 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".