cap p~Pat~J.: l 1
harmonic functions
Overview
A wide variety of problems in engineering and physics involve harmonic func-
tions, which are the real or imaginary part of an analytic function. The stan-
dard 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 pre-
scribed boundary values. Noteworthy methods include Poisson's integral for-
mulae; the Joukowski transformation; and Schwarz-Christoffel transformation.
Modern computer software is capable of implementing these complex analysis
methods.
11. 1 Preliminaries
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 2.4, 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) t hat is harmonic in the vertical
strip a~ Re (z) ~band takes on the boundary values
u(a,y)= U 1 and u(b,y)=U2
along the vertical lines x = a and x = b, respectively.
Solution Intuition suggests that we should seek a solution that takes on con-
stant values along the vertical lines of the form x = x 0 and that u (x, y) be a
function of x alone; that is,
u(x,y) = P(x), for a~ x ~ b and for ally.
Laplace's equation, Uxx (x, y) + u 11 y (x, y) = 0, implies that P" (x) = 0, which
implies P (x) = mx + c, where m and c are constants. The stated boundary
conditions u (a, y) = P (a)= U1 and u (b,y) = P (b) = U2 lead to the solution
U2- U1
u(x, y)=U 1 + b (x-a).
- a
The level curves u ( x, y) = constant are vertical lines as indicated in Figure 11.1.
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