Using Wirtinger calculus and holomorphic matching to obtain the discharge potential for an elliptical pond

We present in this paper a new method for deriving discharge potentials for groundwater flow. Discharge potentials are two-dimensional functions; the discharge potential to be presented represents steady groundwater flow with an elliptical pond of constant rate of extraction or infiltration. The method relies on Wirtinger calculus. We demonstrate that it is possible, in principle, to construct a holomorphic function Ω(z)defined so as to produce the same gradient vector in two dimensions as that obtained from an arbitrary function F(x, y) along any Jordan curve C. We will call Ω(z) the holomorphic match of F{x, y) along C. Let the line C be a closed contour bounding a domain V, and let F(x, y) be defined in V and represent the discharge potential for some case of divergent groundwater flow. Holomorphic matching makes it possible to create a function Ω(z), valid outside V, such that RΩ equals F(x, y) and the gradient of RΩ equals that of F(x, y) along C. (Note that the technique applies also if V is the domain outside C.) We can use this technique to construct solutions for cases of flow where there is nonzero divergence (due to infiltration or leakage, for example) in V but zero divergence outside V. The special case that the divergence within V is constant and is zero outside V is chosen to illustrate the approach and to obtain a solution that, to the knowledge of the author, does not exist in the field of groundwater flow.