User:DVD206/Hilbert transfrom

The Hilbert transform gives a correspondence between boundary values of harmonic function and its harmonic conjugate.

$H:u|_{\partial \Omega }\rightarrow v|_{\partial \Omega },$

where

$f(z)=u(z)+iv(z)$

is an analytic function in the domain.

Exercise (*). Prove that for the case of the complex half-plane C⁺ the Hilbert transform is given by the following formula:

$H_{C^{+}}f(y)={\frac {1}{\pi }}\ {\text{p.v.}}\int _{-\infty }^{\infty }{\frac {f(x)}{y-x}}dx.$

Exercise (*). Differentiate under integral sign the formula above to obtain the kernel representation for the Dirichlet-to-Neumann operator for the uniform half plane.

To define discrete Hilbert transform for a planar network, we need to consider it together w/its dual.