Their Application To Mathematical Physics N I Muskhelishvili - Singular Integral Equations Boundary Problems Of Function Theory And

[ \kappa = \frac12\pi \left[ \arg G(t) \right]_\Gamma. ]

[ (S\phi)(t_0) := \frac1\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt ] [ \kappa = \frac12\pi \left[ \arg G(t) \right]_\Gamma

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 ): \textP.V. \int_\Gamma \frac\phi(t)t-t_0

[ 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, ] \textP.V. \int_\Gamma \frac\phi(\tau)\tau-t

[ \Phi^\pm(t_0) = \pm \frac12 \phi(t_0) + \frac12\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt, ]

with given Hölder-continuous ( G(t) \neq 0 ) and ( g(t) ). The of the problem is