Zeros and Poles of the Functions with Weak Derivatives

The classic results of Gergen J. J., Dressel F. G. are generalized to the class of the functions with weak derivatives. We suppose that these derivatives could be estimated by the proper functions multiplied by the weighted functions which have singularities at isolated boundary points. The crucial point of the study is the iteration process used for the evaluations of the functions represented by the potential operators. As a result of such iterations we succeed in lowering the degree of kernel singularities of the potential operators. The above mentioned method is based on representation formula of I.N. Vekua for the functions whose weak derivatives are summable over domains. The analytic functions participating in these representations could be considered as generalized constants. We study the classes of those functions whose generalized constants have finite numbers of poles and zeros. We prove theorems on behavior of the above mentioned functions in neighborhood of their zeros. Besides we study these functions in the neighborhood of the points where they haven’t finite limits. The main result of the paper is the theorem on the number of zeros and poles of the functions under consideration. This result is the generalization of theorem from the paper of Gergen J. J., Dressel F. G.

weak derivatives, integral representation of functions, zeros and poles of functions, iterated estimates, weighted functions