Debye's asymptotic series is frequently used for calculation of cylindrical functions. However, it seems that until now this series has not been used in all-purpose programs for numerical calculation of the cylindrical functions. The authors attempt to develop these all-purpose programs. We present some improvements for the numerical calculation. As the results, Debye's series can be used for the all-purpose programs, and it is found out that the series gives sufficient accuracy if some conditions are satisfied.

The method solving Bessel's differential equation for calculating numerical values of the Bessel function Jν(x) is not usually used, but it is made clear here that the differential equation method can give very precise numerical values of Jν(x), and is very effective if we do not mind computing time. Here we improved the differential equation method by using a new transformation of Jν(x). This letter also shows a method of evaluating the errors of Jν(x) calculated by this method. The recurrence method is used for calculating the Bessel function Jν(x) numerically. The convergence of the solutions in this method, however, is not yet examined for all of the values of the complex ν and the real x. By using the differential equation method, this letter will numerically ascertain the convergence of the solutions and the precision of Jν(x) calculated by the recurrence method.

There are so many methods of calculating the cylindrical function Zν(x), but it seems that there is no method of calculating Zν(x) in the region of νx and |ν|»1 with high accuracy. The asymptotic series presented by Watson, et al. are frequently used for the numerical calculation of cylindrical function Zν(x) where νx and |ν|»1. However, the function Bm(εx) included in the m'th term of the asymptotic series is known only for m5. 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 Bm(εx) of arbitrary m. We developed a method of calculating Bm(εx) for arbitrary m, and then succeeded in calculating the cylindrical functions in the region νx with high precision.

The fields in the junctions between straight and curved rectangular waveguides are analyzed by using the method of separating variables. This method was succeeded because the authors developed the method of numerical calculation of the cylindrical functions of complex order. As a result, we numerically calculate the reflection and transmission coefficients in the junctions in various situations, and we compare these results with the results by the perturbation method and with the results by Jui-Pang et al.