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Dictionary of Conformal Representations: H. Kober

The present book contains a collection of formulae and prop erties of a number of conformal representations. No proofs are given; in Part IV, however the method of the Schwarz-Christoffel transformation is explained.

Conformal mappings may be classified either according to the analytic functions w = f(z) describing them; or according to the curves and regions which they map, respectively on certain curves and regions in the other plane, for instance on the unit circle and its interior, or on the real axis and the upper half-plane. The first classification can be made more systematic. It has, therefore, on the whole been adopted in the present dictionary. The other method, however, is used throughout Part IV, and also in some sections of the other parts for instance in I, 5 and II, 7.

Completeness in the sense of including all relevant conformal mappings could not be aimed at in this short book, apparently the first of its kind; in which the author tries to deal with the subject in a systematic way and t incidentally, to fill the worst gaps. Alsot in most cases transformations are omitted, if they can be obtained by com- bination of transformations included in this dictionary with linear or bilinear transformations. A typical example at the end of Part I will illustrate the use of the formulae of Part I in combinations. A number of special notations are used in this dictionary.

In the Theory of Functions nobody would use a notation other than |z-z_1| = r for the circle with centre z_1 and radius r; though it is different from the formula familiar in Analytic Geometry. For the straight line, ellipse, etc., however, the forms of Analytic Geometry are usually employed in conformal transformations though, in most cases, they are not suitable for this purpose. By means of the other notations, some of which were used by G. Polya G. Szego, and which are adapted to the present problems, general results, apparently not yet stated, can be obtained in a number of cases e.g. in 3.2, 3.3, 6.1. The notation used for the rectangula hyperbola is due to Professor A. Erdglyi. The present form of the dictionary, aiming at con- ciseness by a proper arrangement of the results, is due to him and to Dr J. Todd. I am greatly indebted to them for this and for other valuable advice. Again I wish to thank Professor L. Rosenhead, Liverpool.

"Written for the British Admiralty during 1944-1948 ... Reproduced and circulated in the Department of Scientific Research and Experiment, and in the Department of Physical Research."

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