Harmonic functions are important in the areas of applied mathematics, engineering, and mathematical physics.  In calculus we might have been asked to show that the functions and   are harmonic (and also that the functions    and    are harmonic).  Harmonic functions are used to solve problems involving steady state temperatures, two-dimensional electrostatics, and ideal fluid flow.  In Chapter 11 we will see how complex analysis techniques can be used to solve some problems involving harmonic functions.  For example, the function    is harmonic in the upper half plane and takes on the boundary values     and   ,  as shown in Figure 3.A.

        Figure 3.A  The harmonic function  .

Exploration

(c) 2006 John H. Mathews, Russell W. Howell