in mathematics, functions belonging to various special classes that are encountered particularly often in solving problems in mathematical physics. The principal special functions are solutions of second-order linear differential equations with variable coefficients. Examples of important special functions are the hypergeometric functions, cylindrical functions, spherical functions, solid spherical harmonics, Lamé functions, and Mathieu functions. Transcendental functions that cannot be expressed in terms of elementary functions are sometimes also called special functions. Such transcendental functions include elliptic functions, the gamma function, the zeta function, the logarithmic integral, and the error function.