- 2 • INVARIANCE OF LAPLACE'S EQUATION AND THE DIRICHLET PROBLEM 433
yFigure ll.7 The boundary values for
connected domain D.
One method for finding <f> is to find a conformal mappingw = f (z) = u(x,y) +iv(x, y) (11-7)
of D onto the upper half-plane Im(w) > 0, such that the N points z1, z2,... , ZN
are mapped onto the points Uk = f (zk), for k = 1, 2, ... , N - 1, and ZN is
mapped onto V.N = +oo along the u-axis in thew plane.
When we use Theorem 11.1, the mapping in Equation (11-7) gives ri se to a
new N -value Dirichlet problem in the upper half-plane Im (w) > 0 for which the
solution is given by Theorem 11.2. If we set Cl-0 = aN, then the solution to the
Dirichlet problem in D with the boundary values from Equation (11-6) is
l N-1
k=I
N - 1 ( )
= aN-1 + .!_ L (ak- 1 - ak) Arctan t x} y.
11' k=l U x,y - Uk
This method relies on our ability to construct a conformal mapping from D
onto the upper half-plane Im (w) > 0. Theorem 10.4 guarantees the existence of
such a conformal mapping.
•EXAMPLE 11 .8 Find a function
lzl < 1 and takes on the boundary values
ef>(x, y) = 0,
ef>(x, y) = 1,
for x + iy = ei^11 , 0 < B < 11';
for x + iy = e;^8 , 11' < B < 271'.
Solution Example 10 .3 showed that the function
. i (1 - z) 2y. 1 - x^2 - y^2
u+iv= = +i-----
1 + z (x+1)^2 +y2 (x+1)^2 +y2
(11-8)(11-9)