defines two analytic functions: ( \Phi^+(z) ) inside, ( \Phi^-(z) ) outside. Their boundary values on ( \Gamma ) satisfy
[ (a(t) + b(t)) \Phi^+(t) - (a(t) - b(t)) \Phi^-(t) = f(t). ] defines two analytic functions: ( \Phi^+(z) ) inside,
where P.V. denotes the Cauchy principal value. The singular integral operator defines two analytic functions: ( \Phi^+(z) ) inside,
[ \Phi(z) = \frac12\pi i \int_\Gamma \frac\phi(t)t-z , dt ] defines two analytic functions: ( \Phi^+(z) ) inside,