434 CHAPTER, 11 • APPLICATIONS OF HARMONIC FUNCTIONS
y- I
w= i(l-z)
l+zFigure 11.8 The Dirichlet problems for lzl < 1 and Im ( w) > 0.vuis a one-to-one conformal mapping of the unit disk izl < 1 onto the upper half-
plane Im (w) > 0. Equation (11-9) reveals that the points z = x + iy lying on
the upper semicircle y > O, 1 - x^2 - y^2 = O are mapped onto the positive u-axis.
Similarly, the lower semicircle is mapped onto the negative u-axis, as shown in
F igure 11.8. The mapping given by Equation (11-9) gives rise to a new Dirichlet
problem of finding a harmonic function <I> (u, v) that has the boundary values(u, 0) = 0, for u > 0, and (u, O) = 1, for u < 0,
as shown in Figure 11.8. Using the result of Example 11.5 and the functions u
and v from Equation {11-9), we get the solution to Equation (11-8):1 v(x,y) 1 1 - x^2 - y^2
<f>(x, y) = - Arctan-(--) = - Arctan
2
.
-rr ux,y -rr y- EXAMPLE 11 .9 Find a function
(x, y) that is harmonic in the upper half-
d isk H : y > 0, lzl < 1 and takes on the boundary values(x, y) = 0,
</> (x, 0) = 1,
for x + iy = ei^9 , 0 < fJ < -rr;
for - 1<x<1.
Solution When we use the result of Exercise 4, Section 10 .2, the function in
Equation (11-9) maps the upper half-disk H onto the first quadrant Q : u >
0, v > 0. The conformal mapping given in Equation (11-9) maps the points
z = x + iy that lie on the segment y = 0, - 1 < x < 1, onto the positive v-axis.
Equation (11-9) g ives rise to a new Dirichlet problem of finding a harmonic
function <I> ( u , v ) in Q that has the boundary values