1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11.2 • lNVARJANCE OF LAPLACE' S EQUATION AND THE DIRICHLET PROBLEM 435

y

-1 ¢(x,0)=1 for-l<x<l

v

w=-i ( I --z)

1 +z

<1>(0, v) = I ,

for v>O

0 cP(u, 0) = 0 for u > 0

F igure 11.9 T he Dirichlet problems for the domains H and Q,

u

as shown in Figure 11.9. In this case, the method in Example 11.2 can be used
to show that iI> ( u, v) is given by

1-0 2 2 v

iI> (u, 0) = 0 + li!Argw = -Argw = -Arctan-.

2 7r 'Ir u

Using the functions u and v in Equation (11-9) in the preceding equation, we
find the solution of the Dirichlet problem in H :

2 v (x y) 2 1 - x^2 - y^2
tj>(x, y) = -Arctan ( ' ) = - Arctan
7r u x, y 7r 2 y
A three-d imensional graph u = 4>(x,y) in cylindrical coordinates is shown in
Figure 11 .10.

• EXAMPLE 11. 10 Find a function cf> (x, y) that is harmonic in the quarter-
  disk G : x > 0, y > 0, lzl < 1 and takes on the boundary values

4>(x,y)=O, forx+iy=z=e^111 , 0<9<~;

4>(x, O) = 1,

4>(0, y) = 1,

for 0 $ x < 1;

for 0 $ y < 1.

Solution The function

u + iv = z^2 = x^2 - y^2 + i2xy (11-10)

maps the quarter-disk onto the upper half-disk H : v > 0, lwl < 1. The new
Dirichlet problem in H is shown in Figure 11.11. From the result of Example
11.9 the solution iI> ( u, v) in H is
2 1 - u^2 - v^2
iI> (u, v) = - Arctan
'Ir 2 v
(11-11)