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.