Abstract: In the paper, special functions frequently encountered in science and engineering are introduced. The computation of the values of Bessel function and elliptic integrals are taken to be the examples, and some common algorithms for computing most special functions, such as series expansion for small argument, asymptotic approximations for large argument, polynomial approximations, recurrence formulas and iteration method, are discussed. In addition, the determination of zeros of some special functions, and the other questions related to numerical computation are also discussed.

Key words: special function, series expansion, asymptotic approximation, recurrence formula