1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

12.2 • THE DIRICHLET PROBLEM FOR THE UNIT DISK 527

s

s

x

0

s = U(I)

Figure 12.13 Functions U1 (t) and u1 (1·cos8,rsin8).

Using Equation (12-11) for the solution of the Dirichlet problem, we obtain

(^00) (-1t+l
u(rcos8,rsin8) = L rnsinnB.
n=l n
This series representation of u(rcos8,rsin8) takes on the prescribed bound-
ary values at points where U (8) is continuous. The boundary function U (8)

is discontinuous at z = -1, which corresponds to 8 = ±11'; U (8) was not

prescribed at these points. Graphs of the selected approximations U 7 (t) and
u1 (x, y) = u1 (rcos8, rsin8), which involve the first seven terms in the preced-
ing two equations, a.re shown in Figure 12.13.

-------.-EXERCISES FOR SECTION 12 .2

For Exercises 1-6, find the solution to the given Dirichlet problem in the unit

disk D by using the Fourier series representations for the boundary functions

that were derived in the examples and exercises of Section 12.1.

1. U(O) = {
   1, for 0 < () < 11';
   
   
   • 1, for - 11' < 8 < 0.
   
   
   
   


2. U(8) = { !J-8, for^0 ~ () < 11'j
   ~ + 8, for - 11' <^8 < O.

Approximations for Ud8) andu5(rcos8,rsinll) are shown in Figure 12.14.