Mathematics of Physics and Engineering

Conformal Mappings 205

real part of w. Since the collection of all w with the same imaginary part
is a line parallel to the real axis, we conclude that the function f(z) = \jz
maps a circle \z — ic\^2 — c^2 to the line {w : Q(w) = -l/(2c)}.

EXERCISE 4.2.26F (a) Verify that the family of circles \z - c\^2 = c^2 , c> 0,
c ^ 0 is orthogonal to the family of circles \z — ic\^2 = c^2 , c £ WL, c > 0
(draw a picture), (b) Verify that the function f(z) = \/z maps a circle
\z — c\^2 = c^2 to the line line 3?w = l/(2c). Again, draw a picture and
convince yourself that all the right angles stayed right, (c) What happens if
we allow c < 0?

For more examples of conformal mappings, see Problem 5.4, page 434.
The following theorem is one of the main tools in the application of
complex analysis to the study of Laplace's equation in two dimensions.

Theorem 4.2.6 Let G and G be two domains in M^2 and let f(z) —
u(x, y) + iv(x, y) be a conformal mapping of G onto G. IfU = U(£, n) is a
harmonic function in G, then the function U(x,y) = U(u(x,y), v(x, y)) is
a harmonic function in G.

The domain G is often either the upper half-plane or the unit disk with
center at the origin. Note the direction of the mapping: for example, to
find a harmonic function in a domain G given a harmonic function in the
unit disk, we need a conformal mapping of the domain G onto the unit disk.

EXERCISE 4.2.27? (a) Prove Theorem 4-2.6 in two ways: (i) by interpreting
U as the real part of analytic function F{f{z)), where F is an analytic
function with real part U; (b) by showing, with the help of the Cauchy-
Riemann equations, that

uxx + um = (UK + £/„„) (u^2 + u^2 y).

(b) Suppose you can solve the Dirichlet problem V^2 U = 0, U\dD = 9, for

every continuous function g when D is the unit disk. Let G be a bounded

domain with a smooth boundary dG, and f : G —» D, a conformal mapping

of G onto D. How to solve the Dirichlet problem V^2 V = 0, V\QG = h, for

a given continuous function h?