A Class of Nonlinear Partial Differential Equations and Their Properties

1. The class Q of complex solutions of a linear partial differential equation. Any two real harmonic functions U(x, y), V(x, y)—that is, solutions of the Laplace equation—can be combined to form a complex harmonic function U+iV. The class of all complex harmonic functions is of no interest, because in effect it possesses no special properties not already possessed by real harmonic functions. However, the theory of analytic functions of a complex variable, which is the subclass of complex harmonic functions where U and V satisfy the Cauchy-Riemann equations, has proved to be a powerful means for the study of real harmonic functions. There is an analogous situation in the case of real solutions of the general linear equation of elliptic type,