Commonly Used Special Functions: Definitions and Computing

Special functions are dedicated mathematical functions invented by mathematicians. For instance, if the integrand is \(f(x)=\textrm{e}^{-x^2}\) , the analytical solution to its indefinite integralsIndefinite integral does not exist. Therefore, mathematicians invented a special function \(\textrm{erf}(x)\) to express the integral expression. This special function is regarded as the analytical solution of the integral problem. In different applications, many other such special functions are invented. For instance, Gamma function and Beta function and so on. Many special functions are invented for indefinite integral and differential equation problems.