Abstract
Gaier’s variational method is used to solve two extremal problems in the theory of conformal mapping. The first deals with the conformal mapping of a simply connected region onto a disk, and the second with that of the boundary of the region onto the circumference of the disk. Both problems use the Ritz method for approximating the minimal mapping function by polynomials. This mapping function in the first problem is represented in terms of the Bergman kernel function, and in the second problem in terms of the Szegö kernel. Another important problem deals with an investigation into the nature and location of boundary singulaities and poles of the mapping function close to the boundary, which is presented in Chapter 12.
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