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

where

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:

**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.