Math 6420 • PDE

Suppose that \(\psi _+(z)\) is analytic on the upper half plane and \(\psi _-(z)\) is analytic on the lower half plane such that on the real axis \[ \psi _+(x)-\psi _-(x)=f(x) \] Assuming that the integrable function \(f(x)\) can be analytically extended in a neighborhood of the real axis, show that a solution to this problem is given by the Cauchy integral \[ \Psi (z)=\frac {1}{2\pi i}\int _{-\infty }^{\infty }\frac {f(\xi )}{\xi -z}\,d\xi \] with \(\psi _\pm (x)=\lim _{\epsilon \to 0+}\Psi (x\pm i\epsilon )\). Hence, establish that \[ \psi _+(x)+\psi _-(x)=\frac {1}{\pi i}\operatorname {PV} \int _{-\infty }^{\infty }\frac {f(\xi )}{\xi -x}\,d\xi \] where \(\operatorname {PV}\) denotes the principal value.