1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

432 CHAPTER 1 1 • APPLICATIONS OF HARMONIC FUNCTIONS

1.0

0.8

0.6

Figure 11.6 The graph of u = <P (x, y) with the boundary values 4> (x, O) = 1, for

lxl < 1, and 4> (x, 0) = 0, for lxl > 1.

Solution This three-value Dirichlet problem has ao = 0, a 1 = 1, and % = 0

and x 1 = - 1 and x 2 = 1. Applying Equation (11-5) yields

0-1 1-0

<P(x , y) = 0 + -Arg (z + 1) + --Arg(z - 1)

'Tr 'Ir

= -l Arctan- Y- + ~Arctan-Y-.

7r x + l 7r x-1

A three-dimensional graph of u = <f>(x, y) is shown in Figure 11.6.

We now state the N -value Di.richlet problem for a simply connected domain.

We let D be a simply connected domain bounded by the simple closed contour C

and let z 1 , z2, ... , ZN denote N points that lie a long C in this specified order as

C is traversed in the positive direction (counterclockwise). Then we let Ck denote

the portion of C that lies strictly between Zk and Zk+l> for k = 1, 2, ... , N - 1,

and let CN denote the portion that lies strictly between Z N and z1. Finally, we

let a1, az,... , aN be real constants. We want to find a function <P (x, y) that

is harmonic in D and continuous on D U C 1 U C2 U · · · U CN that takes on the

boundary values

<f>(x, y) = a1,

<P (x, y) = a2,

<P(x, y) = aN,

for z = x + iy on C1;

for z = x + iy on C2;

for z = x + iy on CN-

The situation is illustrated in Figure 11.7.

(11-6)