1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

436 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

1.0

0.8

y

Figure 11.10

2 1 -x^2 -y^2 2 l-r^2

The graph u = - Arctan = - Arctan

2

.

8

.

1T 2y 1T rsm

?(0,y)= l

for OSy< I

y

¢(x,0)= 1 for O<x<I I

x

-1 4>(u, 0) = I for - I < u < I

Figure 11.11 The Dirichlet problems for the domains G and H.

Using Equation (11-10), we can show that u^2 + v^2 = (x^2 + y^2 )

2

and 2v = 4xy,

which we use in Equation ( 11 -11) to construct the solution</> in G:

2 1 - (x2 +yz)2

</>(x, y) = -Arctan-~--~

1T 4xy

A three-dimensional graph u = <f>(x,y) in cylindrical coordinates is shown in

F igure 11.12.

u