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.

First, the necessity of examining the numerical calculation of the Bessel function Jν(x) of complex order ν is explained. Second, the possibility of the numerical calculation of Jν(x) of arbitrary complex order ν by the use of the recurrence formula is ascertained. The rounding error of Jν(x) calculated by this method is investigated next by means of theory and numerical experiments when the upper limit of recurrence is sufficiently large. As a result, it was known that there is the possibility that the rounding error grows considerably when ν is complex. Counterplans against the growth of the rounding error will be described.

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.

Windows in rectangular waveguides are used frequently in microwave circuits. Capacitive or inductive windows have been studied in detail. Though the problems for the fields of these windows can be reduced to one-dimensional problems, the field of the resonance window is obtained by solving a two-dimensional problem. Thus the field of the resonance window is more difficult to obtain. Various characteristics of the resonance window in the rectangular waveguide have not been much studied analytically. A simple and approximate expression which was obtained experimentally for calculating the resonance frequency of the window is well known. The authors study the various characteristics of the resonance window by the variational method and Schwinger's transformation. We show an expression for susceptance of the window by these methods, and we can evaluate the resonance frequency and loaded Q numerically. These data by the numerical calculation are shown in figures, which may be useful to the design of microwave circuits. These data are compared to experimental data and theoretical values for windows of special cases. As a result, it is found that the data illustrated with the figures hold sufficient accuracy.

The characteristics of an echelette grating are analyzed by the region dividing method which divides the free-space region above the grating into some subregions which overlap one another. The field in each subregion is expressed by the method of separation of variables; the coefficients of the modal functions in each subregion are determined by using the orthogonality of these functions. Hence, this method eliminates many of the unknown coefficients analytically, and gives sufficiently exact characteristics without regard to the shape of the echelette grating. One of the subregions above the grating is a sector. This paper shows the method of separation of variables applied to the sectorial region; a new expansion for an arbitrary function by orthogonal functions composed of the cylindrical functions is derived. The properties of the cylindrical function used in this expansion and a new method of numerical calculation of the function are given.

The FDTD method needs Fourier analysis to obtain the fields of a single frequency. Furthermore, the frequency spectra of the fields used in the FDTD method ordinarily have wide bands, and all the fields in FDTD are treated as real numbers. Therefore, if the permittivity ε and the permeability µ of the medium depend on frequency, or if the surface impedance used for the surface impedance boundary condition (SIBC) depends on the frequency, the FDTD method becomes very complicated because of convolution integral. In the electromagnetic theory, we usually assume that the fields oscillate sinusoidally, and that the fields and ε and µ are complex numbers. The benefit of introduction of the complex numbers is very extensive. As we do in the usual electromagnetic theory, the authors assume that the fields in FDTD oscillate sinusoidally. In the proposed FDTD, the fields, ε, µ and the surface impedances for SIBC are all treated as the complex numbers. The proposed FDTD method can remove the above-mentioned weak points of the conventional FDTD method.

Cutoff frequencies and the modal fields in hollow conducting waveguides of arbitrary cross section are frequently calculated by the method of solving integral equations. This paper presents some improvements for the method by the integral equations. The improved method can calculate the cutoff frequencies and the modal fields only by using the real number, and this method can remove extraneous roots when calculating the cutoff frequencies. The former method calculates the cutoff frequency and the fields only at the cutoff frequency, but the improved method can calculate the fields at arbitrary phase constants.

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.

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.

When we study time-domain electromagnetic fields, we frequently use the finite-difference time-domain (FD-TD) method. In this paper, we discuss errors of the FD-TD method and present the optimum mesh spacings in the FD-TD method when the three mesh spacings are different.

When we use the finite-difference time-domain (FD-TD) method to study time-domain electromagnetic fields in the unbounded surroundings, we frequently use a radiation boundary condition (RBC) by means of one-way wave equations. The reflection coefficient by the RBC is independent of frequency, but the reflection coefficient of the finite difference approximation for the RBC depends on a frequency also; this study examines how the reflection characteristics are affected by the frequency, and the study presents the coefficients used in the RBC which gives expected reflection characteristics for a frequency, and presents the application to simulation of the matched termination of a rectangular waveguide.

We frequently use a polynomial to approximate a complex function. This study shows a method which determines the optimum coefficients and the number of terms of the polynomial, and the error of the polynomial is estimated.

This letter proposes an improvement of the equivalent source method in order to give an accurate solution for the scattering of an electromagnetic plane wave by a conducting cylinder with edges.

Generalized telegraphist's equations for waveguides are frequently used for analyzing fields in tapered rectangular waveguides. However, the telegraphist's equations do not give exact fields in E-plane tapers of the rectangular waveguides. In this letter, the new telegraphist's equations are shown, and the equations give the fields which exactly satisfy Maxwell's equations and the boundary conditions for the E-plane taper of the waveguide.

New series expressing the radiation fields from both axial and circumferential slots on a circular conducting cylinder are derived. These new series converge rapidly even for near fields. This letter includes useful figures showing characteristics of near fields calculated numerically using the new series.

Scattering of a plane electromagnetic wave by a conducting wedge will be discussed. The former solution can not be applicable to all the transition regions when its parameter is constant. This study shows a new solution which consists of only one expression applicable to the shadow region, the illuminated region and the transition regions, and which has no parameter.

Calculation Nv(x) of complex order v numerically, we must calculate Df{JN+ε(x)}. When Df{JN+ε(x)} is calculated by the recurrence method, this letter will analyze the error of Df{JN+ε(x)}, and will determine the optimum number of recurrences.

The recurrence method is useful for numerical calculation of the Bassel function Jv(x) of complex order v. The necessary total number of the recurrences in this method has been examined for the real order v, but it is known only for limited ranges of the real order v and the variable x, and it is not known for the complex order v. This letter proposes a new algorithm which increases the total number of the recurrences gradually, and which stops the calculation automatically when the approximate Bessel function with a necessary precision is obtained.

In order to study reflection characteristics of a tapered waveguide, we often use an equivalent transmission line satisfying the telegraph equations. We show a method of determining the telegraph equation in tapered waveguides having two symmetric planes.