Besides an extensive study of elliptic functions and integrals, classical complex analysis was heavily occupied with solving Laplace's equation in one form or another. Several configurations were sufficiently symmetrical that it was possible to solve the equation by separation of variables, leading to a variety of second order real differential equations. The involvement of complex analysis was a usual one, given that real functions inherit many of their properties from the complex plane, particularly through the location of singularities therein.

With the advent of quantum mechanics, Schrödinger's equation took the form of Laplace's equation with a potential term, so that most of the same functions were still useful, even if they appeared in a somewhat different form or with different parameters.

Each of the special cases has an extensive literature which it is unnecessary to repeat. However it is worth listing the principal functions, with an in indication of where they arose and why they are interesting. Some illustrtate general points and so are worth disscussing in more detail. Having originated from separating variables in a partial differential equation, they all present themselves as eigenvalue equations. Analytic continuation of eigenvalues, parameters, or both, is often useful, aside from the already informative consequence of complexifying real functions.

- survey of equations
- Bessel functions
- Legendre polynomials
- Mathieu functions
- Airy functions for a linear potential
- Dirac harmonic oscillator