Each harmonic function of x and y is a real or imaginary part of some function f(ξ), and conversely, the real and imaginary parts of any analytic function are harmonic functions of x and y. The latter are closely related to the analytic functions f(ξ) of a complex variable ξ = x + iy.
In a particular case, when the region of space is bounded by a cylindrical surface whose generatrices are parallel-for example, the z axes-the phenomenon under investigation proceeds in the same manner in any plane perpendicular to the generatrices (it does not depend on the z coordinate), the corresponding harmonic functions of three variables are converted into harmonic functions of two variables, x and y. Harmonic functions of three variables (the coordinates of a point) are the most important for use in physics and mechanics. The potential of the velocities of a steady-state irrotational motion of an incompressible fluid, the temperature of a body under conditions of a steady-state heat distribution, and the bending of a membrane stretched over an arbitrary contour that is completely nonplanar (the membrane’s weight is ignored) are also harmonic functions. For example, the potential of the gravitational forces in a region that contains no attracting masses and the potential of a constant electrical field in a region containing no electrical charges are harmonic functions. In many problems of physics and mechanics concerned with the state of a portion of space, which depends on the position of a point but not on time (equilibrium, steady-state motion, and so on), the corresponding state is represented by a harmonic function of the point’s coordinates. Functions of n variables ( n≤2) that are continuous over some domain together with their partial derivatives of the first and second orders and that satisfy Laplace’s differential equation in this domain: