I had recently ended up with a case where on solving the Laplace equation (for a fluid under certain conditions), the radial dependence turned out to be complex (in general). In such cases, do we work with the real part only? Or should we not neglect the complex part and I am wrong? I can post the exact problem if needed, but this stands as a general question as well. Any hints/tips would be appreciated.
In a truly general case, complex solutions are not only possible, but useful. For a 2D, incompressible, irrotational flow, there are two useful functions related to the velocity: the velocity potential and the stream function. Both functions reduce a system of equations for the velocity into a scalar equation of higher order. This results in a Laplace equation.
We can, of course, solve for either the velocity potential or the stream function. But, we can solve for both at the same time, where the velocity potential is the real component of the solution to the Laplace equation, and the stream function is the complex component of the solution to the Laplace equation. See this page for more details