**for**

**Chapter 11 Applications of
Harmonic Functions**

**11.1 Preliminaries**

**Overview**

A wide variety of problems in engineering and
physics involve harmonic functions, which are the real or imaginary
part of an analytic function. The standard applications
are two dimensional steady state temperatures, electrostatics, fluid
flow and complex potentials. The techniques of conformal
mapping and integral representation can be used to construct a
harmonic function with prescribed boundary
values. Noteworthy methods include Poisson's integral
formulae; the Joukowski transformation; and Schwarz-Christoffel
transformation. Modern computer software is capable of
implemeting these complex analysis methods.

In most applications involving harmonic functions, a harmonic function that takes on prescribed values along certain contours must be found. In presenting the material in this chapter, we assume that you are familiar with the material covered in Sections 2.5, 3.3, 5.1, and 5.2. If you aren't, please review it before proceeding.

**Example 11.1.** Find
the function u(x,y) that is harmonic
in the vertical strip
and takes on the boundary values

for
all y, and

for
all y,

along the vertical lines , respectively.

Solution. Intuition suggests that we should seek a
solution that takes on constant values along the vertical lines of
the form
and that u(x,y) be a function of
x alone; that is,

, for and
for all y.

Laplace's equation, , implies
that , which
implies , where
m and c
are constants. The stated boundary
conditions and lead
to the solution

.

The level curves are
vertical lines as indicated in Figure 11.1.

Level curves of the harmonic function .Figure 11.1

**Example 11.2.** Find
the function
that is harmonic in the sector
and takes on the boundary values

for x
> 0,

for
all points on the ray .

Solution. Recalling that the
function is
harmonic and takes on constant values along rays emanating from the
origin, we see that a solution has the form

,

where a and b
are constants. The boundary conditions lead
to

.

The level curves are
rays emanating from the origin as indicated in Figure 11.2.

Level curves of the harmonic function .Figure 11.2

**Example 11.3.** Find
the function
that is harmonic in the annulus and
takes on the boundary values

when , and

when .

Solution. This problem is a companion to the one in
Example 11.2. Here we use the fact
that is
a harmonic function, for all . The
solution is

,

and the level curves are
concentric circles, as illustrated in Figure 11.3.

Level curves of the harmonic function .Figure 11.3

**Exercises
for Section
11.1 Preliminaries****
**

**The Next Module
is**

**Laplace's
Equation and Dirichlet Problem **

**Return to the Complex
Analysis Modules **

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell