Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics (Dover Books on Physics)

Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics (Dover Books on Physics)
N. I. Muskhelishvili | 2008-05-19 00:00:00 | Dover Publications | 464 | Mathematics

This high-level treatment by a noted mathematician considers one-dimensional singular integral equations involving Cauchy principal values. Intended for graduate students and professionals, its coverage includes such topics as the Hölder condition, Hilbert and Riemann-Hilbert problems, the Dirichlet problem, inversion formulas for arcs, and many other areas. 1992 edition.

Reviews
To be honest, I have not gone into a thourough and complete reading of the texbook, but I picked up several and sparse subjects in order to get a better understanding about some issues relating linear transport theory. But also in a limited use like that I cannot help recognizing that, even after half a century, this book remains a milestone in integral equations and boundary problems. Deep, clear, and available also to ready-to-use requests, it may seem a bit out of fashion for the light use of geometrical formalism, but also this latter feature makes it to be very strict to the point, if you need to strengthen the theoretical building of numerical calcualtions to be performed.
Reviews
Eventhough this text dates from 1946 it is difficult to find a more complete and accurate discussion on the boundary value problems raised up by the Cauchy integral.

Being one of the most outstanding pupils of the Vekua school (as Gakhov), Muskhelishvili explores every single particular case of the classical boundary value problems of complex analysis giving a complete solution to each, sometimes employing highly ingenious arguments. It includes also several applications.

As in the case of the book by Gakhov (also reviewed by myself) I wonder why this material is not standard in usual complex analysis courses, at least in the American continent. When you read this kind of books you realize that all your previous knowledge on the subject was almost useless, to say the least. My suggestion for the material to be mastered by complex analysis students is: First read an introductory classic like Ahlfors, Lang, or Markushevitch, and then proceed to Kress (Linear Integral Equations), Gakhov, and Muskhelishvili. Then you will be ready for the next step: Hypercomplex analysis.

The contents of the book are: The Hölder condition; Integrals of the Cauchy type; Some corollaries on Cauchy integrals; Cauchy integrals near the ends of the line of integration; The Hilbert and Riemann-Hilbert boundary problems; Singular integral equations with Cauchy kernels (case of contours); The Dirichlet problem; Various representations of holomorphic functions by Cauchy and analogous integrals; Solution of the generalized Riemann-Hilbert-Poincaré problem; The Hilbert problem in the case of arcs or discontinuous boundary conditions; Inversion formulae for arcs; Effective solution of some boundary problems of the theory of harmonic functions; Effective solution of the principal problems of the static theory of elasticity for the half-plane, circle and analogous regions; Singular integral equations for the case of arcs and continuous coefficients; Singular integral equations in the case of discontinuous coefficients; Application to the Dirichlet problem and similar problems; Solution of integro-differential equations of the theory of aircraft wings of finite span; The Hilbert problem for several unknown functions; Systems of singular integral equations with Cauchy type kernels and some supplements; + 3 appendices.

Includes full motivation for each topic, historical notes, and extensive references.

Please read some of my other reviews (just click on my name above).

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