[ a(t) \phi(t) + \fracb(t)\pi i , \textP.V. \int_\Gamma \frac\phi(\tau)\tau-t , d\tau = f(t), \quad t \in \Gamma, ]
with given Hölder-continuous ( G(t) \neq 0 ) and ( g(t) ). The of the problem is
This is a foundational text in analytical methods for applied mathematics, elasticity, and potential theory. It systematically develops the theory of using the apparatus of boundary value problems of analytic functions (Riemann–Hilbert and Hilbert problems). Core Mathematical Content 1. Prerequisite: Cauchy-Type Integrals and the Plemelj–Sokhotski Formulas Let ( \Gamma ) be a smooth or piecewise-smooth closed contour in the complex plane (often the real axis or a circle). For a Hölder-continuous function ( \phi(t) ) on ( \Gamma ), the Cauchy-type integral
with ( a(t), b(t) ) Hölder continuous. The key is to set [ a(t) \phi(t) + \fracb(t)\pi i , \textP
[ \Phi^+(t) = G(t) , \Phi^-(t) + g(t), ]
is bounded on Hölder spaces and ( L^p ) ((1<p<\infty)). Find a sectionally analytic function ( \Phi(z) ) (vanishing at infinity as ( O(1/z) ) for the “exterior” problem) satisfying on ( \Gamma ):
[ \Phi^\pm(t_0) = \pm \frac12 \phi(t_0) + \frac12\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt, ] It systematically develops the theory of using the
where P.V. denotes the Cauchy principal value. The singular integral operator
[ (S\phi)(t_0) := \frac1\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt ]
then the boundary values yield:
[ \Phi(z) = \frac12\pi i \int_\Gamma \frac\phi(\tau)\tau-z , d\tau, ]
[ (a(t) + b(t)) \Phi^+(t) - (a(t) - b(t)) \Phi^-(t) = f(t). ]
Title: Singular Integral Equations: Boundary Problems of Function Theory and Their Application to Mathematical Physics Author: N. I. Muskhelishvili (also spelled Muskhelishvili) Original Russian Publication: 1946 (frequently revised) English Translation: 1953 (P. Noordhoff, Groningen; later Dover reprints) For a Hölder-continuous function ( \phi(t) ) on