Measurable physical quantities do not directly involve complex numbers (with possible exceptions in quantum physics). However, since most of the standard and special functions in the applied sciences are analytic functions, both mathematical analysis and computational procedures can benefit greatly from exploiting this feature. While such mathematical tools have seen much use during the last couple of centuries (residue calculus, methods for asymptotic expansions, etc.), the realization is very recent that FD methods, specialized for analytic functions in the complex plane, can be remarkably effective. Since analytic functions (and separately their real and imaginary parts) in singularity-free regions satisfy Laplace’s equation, it can be exploited that FD schemes for such applications need only to be accurate for the very small subset of functions of two variables that satisfy this equation.