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.

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.

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.

Hankel's asymptotic expansions are frequently used for numerical calculation of cylindrical functions of complex order. We beforehand need to estimate the precisions of the cylindrical functions calculated with Hankel's asymptotic expansions in order to use these expansions. This letter presents comparatively simple expressions for rough estimations of the errors of the cylindrical functions calculated with the asymptotic expansions, and features of the errors are discussed.