Note on the Inversion of the Schwarz-Christoffel Conformal Transformation

An inversion procedure, based on the methods used in proving Btirmasm’s theorem, is used to provide an integral expression which exhibits the form of the electrostatic field explicitly in terms of the field coordinates. The method is illustrated with an example of a stepped-guide junction. The form of the field and the expression for the mode expansion coefficients are examined. The results are related to the companion problem of solving for the transverse field from a singular integral equation formulation. The two methods agree in the particular case of a two-to-one step for which special simplifications are possible. In the general case, progress in the solution of a class of double-kernel integral equations may be expected through the indirect use of the inversion of the solution obtained from the conformal transformation methods. I. lNTRODUCTION conformal transformation (S–CCT) [2]. Both methods provide solutions, in the first instance, to a quasi-static order, and both are capable of providing higher order corrections to the basic form of solution. Although they appear to have some features in common, both their approach to a problem and their ranges of useful application are different. The TSIE method provides an explicit formula for the transverse electric field and establishes such parameters as obstacle susceptance, mode conversion coefficients, etc. It applies to a range of rectangular waveguide configurations which includes, for example, the boundary at a magnetized ferrite section, an arrangement which is not, apparently, capable T WO powerful analytic techniques for the solution of being tackled by the S–CCT techniques. The latter of a wide class of waveguide problems are the provides an implicit relation for the electric field comtransverse singular integral equation method ponents from which the waveguide and mode param(TSIE) [1] and the use of the Schwarz-Christoffel eters can be extracted by an indirect process. The two methods overlap in the area of single-opening waveguide diaphragms, where they establish results with about Manuscript received October 26, 1970; revised January ~1, 1971. The author is with the Department of Electrical Engineering, equal ease. One reason why the S–CCT technique is as University of Colorado, Boulder, Colo. 80302. readily applicable here as the integral equation method Authorized licensed use limited to: Princeton University. Downloaded on April 26, 2009 at 12:59 from IEEE Xplore. Restrictions apply. LEWIN : INVSRS1ON OF THE CONFORMAL TRANSFORMATION 543 seems to be that in this particular case the implicit formula for the electric field is capable of straightforward analytical inversion to yield a closed-form explicit solution for the complex potential, and this makes the subsequent operations very much easier [2]. On the other hand, this inversion is not mandatory, and solutions can be found for waveguide parameters by suitable indirect means, namely, asymptotic expansions for large axial distance. In contrast, the TSIE method, which yields, when successful, an explicit form for the electric field, is necessarily limited to those problems for which such an explicit solution in closed form can be obtained. Since, as is well known, the formulas arising from the Schwarz– Christoffel transformation are in general not capable of being inverted to yield the field in explicit form, this would seem to provide a limitation on the TSI E method in that the presence of a noninvertible conformal transformation solution would, by implication, preclude the existence of a TSIE explicit solution. In practice the difficulty shows up in that although the singular integral equation for the problem can be formulated without much trouble, the methods of solution available in the literature on singular integral equations [3] do not apply, and an extension ~of the mathematical techniques is necessary if progress is to be made. Such an extension, in a limited area, was in fact achieved in solving the problem of a waveguide step in which the step ratio was 2 to 1 [1]. The limitation on step ratio was dictated by the need to express a cosine of a multiple of an angle in terms of the cosine of the angle, When the multiple is twice, this can be easily done and the resultant equation turned out to be solvable: the electric field at the junction was ultimately exhibited in explicit form. What happens to the S–CCT equations in this case? It transpires that the implicit relation for the potentials simplifies to such an extent when the step ratio is 2 to 1 that it reduces to an algebraic cubic equation with known solution, so that eventually the electric field can be extracted in explicit form in this special example. In both cases, therefore, a special case exists in which, on the one hand, a novel type of solution to a singular integral equation is found, and on the other the conformal transformation is solvable in closed form for the potentials. It was the confluence of these two exceptional cases in this particular problem that provided the starting point for the present investigation. If an extension of the techniques of solving singular integral equations could be achieved, this would enable some insight to be obtained on the inversion of the conformal mapping solutions, even though the resulting expressions might be in the form of integrals not capable of final expression in terms of existing transcendental functions. On the other hand, if such integrals could be found for the inversion of the conformal mapping solutions, then this might be used to throw some light on the solution of an extended class of singular integral equations. Either way the results might be expected to provide a useful extension of existing techniques. It turns out that the inversion of the Schwarz– Christoffel equations was the easier probllem to tackle. Emphasis on the electric fields rather than the potentials appreciably eased the analysis. The singular integral equation solution appears rather more difficult, and knowing a solution in a particular case has not so far assisted in finding the hoped-for extensions. The present results report mainly on the inversion procedure. II. WAVEGUIDE STEPPED JUNCT [ON: INITIAL FORMULATION Fig. 1 shows a waveguide of internal width b stepped symmetrically at z = O to a width d. If t =z+iy is a complex variable describing points internal to the guide, then the equation [4] dt c ((2 – @l/2 z= F (t’ – 1) ’/2 (1) maps the inner surface of the guide onto the real ~ axis. The further transformation U+; V= W = log r maps the { region to the inside of a parallel-plate region, the total transformation being