Algorithms for the computation of Hankel functions of complex order

Hankel functions of complex order and real argument arise in the study of wave propagation and many other applications. Hankel functions are computed using, for example, Chebyshev expansions, recursion relations and numerical integration of the integral representation. In practice, approximation of these functions is required when the order and the argumentz are large.When andz are large, the Chebyshev series expansion of the Hankel function is of limited use. The situation is remedied by the use of appropriate asymptotic expansions. These expansions are normally expressed in terms of coefficients which are defined recursively involving derivatives and integrals of polynomials. The applicability of these expansions in both numerical and symbolic software is discussed with illustrative examples.