Numerical calculation of cylindrical functions in the transitional regionsusing 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 regi on of nu = +/-x and \ nu \ much greater than 1 with high accuracy. The asym ptotic series presented by Watson, et al. are frequently used for the numer ical 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 t o 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 asymptoti c series. For this purpose, we must know the function B-m (epsilonx) of arb itrary m. We developed a method of calculating B-m (epsilonx) for arbitrary m, and then succeeded in calculating the cylindrical functions in the regi on nu = +/-x with high precision.