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

then the boundary values yield:

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

[ \Phi(z) = \frac12\pi i \int_\Gamma \frac\phi(t)t-z , dt ]

[ \Phi(z) = \frac12\pi i \int_\Gamma \frac\phi(\tau)\tau-z , d\tau, ] then the boundary values yield: defines two analytic

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

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

This becomes a Riemann–Hilbert problem with ( G(t) = \fraca(t)-b(t)a(t)+b(t) ). Solvability and number of linearly independent solutions depend on the index. [ a(t) \phi(t) + \fracb(t)\pi i \int_\Gamma \frac\phi(\tau)\tau-t d\tau + \int_\Gamma k(t,\tau) \phi(\tau) d\tau = f(t), ] The singular integral operator [ \Phi^\pm(t_0) = \pm

[ (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)

with given Hölder-continuous ( G(t) \neq 0 ) and ( g(t) ). The of the problem is d\tau = f(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, ]

[ \Phi^+(t) = G(t) , \Phi^-(t) + g(t), ]

with ( a(t), b(t) ) Hölder continuous. The key is to set