By F.D. Gakhov

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

The emphasis of the ebook is at the answer of singular imperative equations with Cauchy and Hilbert kernels. even supposing the booklet treats the speculation of boundary price difficulties, emphasis is on linear issues of one unknown functionality. The definition of the Cauchy style imperative, examples, restricting values, habit, and its central worth are defined. The Riemann boundary worth challenge is emphasised in contemplating the speculation of boundary worth difficulties of analytic services. The publication then analyzes the appliance of the Riemann boundary worth challenge as utilized to singular imperative equations with Cauchy kernel. A moment basic boundary price challenge of analytic features is the Hilbert challenge with a Hilbert kernel; the appliance of the Hilbert challenge is additionally evaluated. using Sokhotskis formulation for sure necessary research is defined and equations with logarithmic kernels and kernels with a vulnerable strength singularity are solved. The chapters within the ebook all finish with a few old briefs, to offer a history of the problem(s) mentioned.

The publication should be very invaluable to mathematicians, scholars, and professors in complex arithmetic and geometrical services.

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8). 10) which will be frequently employed hereafter. 3. $') 26 BOUNDARY VALUE PROBLEMS be the equation of the contour in the complex form, t(s) being a function of the arc s measured from an arbitrary point of the contour. Substituting into the expression for the function φ(ί) the complex coordinate, and separating the real and imaginary parts we have φ(ί) = (p[í{s)] = φι(5) + iφ2(s)^ It is easy to prove that in general the following question has a negative answer: does there exist a function analytic in the domain i)+(/)-), such that the prescribed complex function φ{ή is its limiting value on the contour?

They are fundamental for all following considerations. Let us now state the deduced result. THEOREM. Let L be a smooth contour {closed or open) and φ {τ) a function of position on the contour, which satisfies the Holder con dition. 8). 10) which will be frequently employed hereafter. 3. $') 26 BOUNDARY VALUE PROBLEMS be the equation of the contour in the complex form, t(s) being a function of the arc s measured from an arbitrary point of the contour. Substituting into the expression for the function φ(ί) the complex coordinate, and separating the real and imaginary parts we have φ(ί) = (p[í{s)] = φι(5) + iφ2(s)^ It is easy to prove that in general the following question has a negative answer: does there exist a function analytic in the domain i)+(/)-), such that the prescribed complex function φ{ή is its limiting value on the contour?

4. Limiting values of the derivatives. Derivatives of limiting values. Derivatives of a singular integral 1. Let φ(ή be a function of position on a closed contour L, themth derivative of which satisfies the Holder condition. 14) L $(»·)-(/) = - ^φ('-)φ + -L Γ ^ ^ d T . 3) that the wth derivative of a Cauchy type integral has the form (z) = ^ Ini i j (T - zT^^ dr, Let us integrate the right-hand side m times by parts. Since the con tour is closed, the integrated part vanishes every time. 15). 2.