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

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

[ \Phi^\pm(t_0) = \pm \frac12 \phi(t_0) + \frac12\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt, ] defines two analytic functions: ( \Phi^+(z) ) inside,

[ \Phi(z) = \frac12\pi i \int_\Gamma \frac\phi(\tau)\tau-z , d\tau, ] \textP.V. \int_\Gamma \frac\phi(t)t-t_0

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