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

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

[ \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 ,

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

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

[ (S\phi)(t_0) := \frac1\pi i , \textP.V. \int_\Gamma \frac\phi(t)t-t_0 , dt ]