Identities in the theory of conformal mapping

Introduction. In the theory of conformal mapping numerous canonical domains are considered upon which a given domain may be mapped. The functions performing this map are functions of the domain considered and might be called domain functions. Numerous relations between domain functions of different types are known; very many of these functions may be constructed from a few fundamental ones, such as Green's and Neumann's functions of the domain and the harmonic measures of the boundary continua. But these fundamental functions themselves are also closely interrelated and permit numerous identities. It is of interest to organize the system of relations between the domain functions into a simple form. This is convenient for the theory of variation of domain functions with their domnain; in fact, we obtain often in extremum problems relative to domain functions several different characterizations of the extremum domain, depending on the type of variation applied in the investigation. It is, therefore, essential to be able to reduce one type of equation to another by means of the various identities for domain functions. An understanding of all identities between domain functions may be obtained by sustained application of Schottky's theory of multiply-connected domains [15](i). Schottky proved that there is a close relation between the mapping theory of these domains and the theory of closed Riemann surfaces; the identities among domain functions have their complete analogue in the theory of Abelian integrals and might be proved by means of the latter. It seems, however, that a theory will be of interest which operates only with concepts of conformal mapping and the geometric properties of the functions considered. The functions which prove in such a theory to be the more basic domain functions may be expected to have importance in the general study of conformal mapping, too. In fact, it will be seen that one of the most fundamental functions in the theory will be a kernel function. This type of function has been studied from various points of view recently [1, 2, 13]. The development in this paper gives further a new understanding of variation formulas which have been applied frequently in conformal mapping; it allows us also to carry out variational meLhods in extremum problems with additional conditions, such as the invariance of conformal type.