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

where P.V. denotes the Cauchy principal value. The singular integral operator

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 ): where P

[ \Phi(z) = \frac12\pi i \int_\Gamma \frac\phi(\tau)\tau-z , d\tau, ] \textP.V. \int_\Gamma \frac\phi(\tau)\tau-t

[ 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, ] d\tau = f(t)

defines two analytic functions: ( \Phi^+(z) ) inside, ( \Phi^-(z) ) outside. Their boundary values on ( \Gamma ) satisfy