By F.D. Gakhov

Boundary price difficulties is a translation from the Russian of lectures given at Kazan and Rostov Universities, facing the idea of boundary price difficulties for analytic services.

The emphasis of the booklet is at the answer of singular vital equations with Cauchy and Hilbert kernels. even though the ebook treats the idea of boundary worth difficulties, emphasis is on linear issues of one unknown functionality. The definition of the Cauchy kind necessary, examples, proscribing values, habit, and its primary price are defined. The Riemann boundary price challenge is emphasised in contemplating the idea of boundary worth difficulties of analytic capabilities. The e-book then analyzes the applying of the Riemann boundary worth challenge as utilized to singular crucial equations with Cauchy kernel. A moment basic boundary price challenge of analytic capabilities is the Hilbert challenge with a Hilbert kernel; the appliance of the Hilbert challenge can be evaluated. using Sokhotskis formulation for sure vital research is defined and equations with logarithmic kernels and kernels with a susceptible energy singularity are solved. The chapters within the publication all finish with a few historic briefs, to provide a heritage of the problem(s) mentioned.

The e-book may be very important to mathematicians, scholars, and professors in complex arithmetic and geometrical capabilities.

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**Sample text**

6. This result may be formulated as follows. THEOREM. If an integral of the Cauchy type is taken over a contour having a finite number of corner points, the limiting values of the integral exist, and for the non-corner points the ordinary Sokhotski for-. 18), ( 4 . 1 9 ) take their place. The proof given above cannot be directly extended to the case of cusps. 18), ( 4 . 1 9 ) remain vahd in this case as well. It is only necessary to set ¿x = 0 or α = 2π, depending on whether the point of the cusp is directed to the right or to the left of the contour.

15) by successive differentiation and integration by parts. 5. The Sokhotski formulae for corner points of a contour In investigating problems of existence of the a singular integral and limiting values of a Cauchy type integral, we everywhere assumed the condition that the integration contour is a smooth hne. It is easily seen that this condition is not necessary. 3) in replacing the quantity \ατ\ = ds by mdr; in both cases the smoothness in the immediate vicinity of the investigated point was relevant.

In order that φ{t) be the boundary value of a func tion analytic in the exterior domain D' and takes at infinity the given value Γ, it is necessary and sufficient that the condition ( 4 . 1 2 ) is satisfied. 28 BOUNDARY VALUE PROBLEMS A more general problem may be formulated, namely: to find the condition that a complex function given on the contour is the bound ary value of a function analytic in the domain, except at some isolated points where it has prescribed singularities. For subsequent con siderations it is of interest to examine the case when φ(ί) is the bound ary value of a function analytic in the domain D", except at infinity where it has a pole of order n, the principal part being known: y(z) = οτ,ζ" + fliz"-i + .