Numerical calculation of cylindrical functions in the transitional regions using asymptotic series
There are so many methods of calculating the cylindrical function Z(nu) (x), but it seems that there is no method of calculating Z(nu) (x) in the region of nu = +/-x and \ nu \ much greater than 1 with high accuracy. The asymptotic series presented by Watson, et al. are frequently used for the numerical calculation of cylindrical function Z(nu) (x) where nu = +/-x and \v \ much greater than 1. However, the function B-m (epsilonx) included in the m'th term of the asymptotic series is known only for m less than or equal to 5. Hence, the asymptotic series can not give sufficiently accurate values of the cylindrical functions. The authors attempt to develop programs for the numerical calculation of the cylindrical functions using this asymptotic series. For this purpose, we must know the function B-m (epsilonx) of arbitrary m. We developed a method of calculating B-m (epsilonx) for arbitrary m, and then succeeded in calculating the cylindrical functions in the region nu = +/-x with high precision.
numerical calculation cylindrical function transitional region asymptotic series