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

[ (a(t) + b(t)) \Phi^+(t) - (a(t) - b(t)) \Phi^-(t) = 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, ]

[ (S\phi)(t_0) := \frac1\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt ] [ (a(t) + b(t)) \Phi^+(t) - (a(t) - b(t)) \Phi^-(t) = f(t)

then the boundary values yield:

defines two analytic functions: ( \Phi^+(z) ) inside, ( \Phi^-(z) ) outside. Their boundary values on ( \Gamma ) satisfy \int_\Gamma \frac\phi(t)t-t_0 , dt ] then the boundary

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

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

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), ]

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

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