Harmonic Conjugate Function

The harmonic conjugate to a given function  is a function  such that

is complex differentiable (i.e., satisfies the Cauchy-Riemann equations). It is given by

where , , and  is a constant of integration.

Note that  is a closed form since  is harmonic, . The line integral is well-defined on a simply connected domain because it is closed. However, on a domain which is not simply connected (such as the punctured disk), the harmonic conjugate may not exist.