Page 786 - Advanced_Engineering_Mathematics o'neil

766 CHAPTER 23 Conformal Mappings and Applications

In theory then, two domains D and D (neither of which is the entire plane) can be mapped
∗
conformally onto each other in a one-to-one fashion. This does not, however, make such a confor-
mal mapping easy to find. Bilinear transformations are conformal and will be suitable for some
domains, but they will not be enough in general. However, here is an observation that is central
to constructing a conformal mapping between two domains.
A conformal mapping f : D → D will map the boundary of D into the boundary of D .
∗
∗
This can be exploited as follows. Suppose D is bounded by a path C (not necessarily closed),
which separates the z-plane into two complementary domains D and D. Similarly, suppose D ∗
∗ ∗ ∗
is bounded by a path C , which separates the w-plane into complementary domains D and D
∗
(Figure 23.18). Try to find a conformal mapping f that sends points of C to points of C .This
may be easier than mapping the entire domains to each other. This mapping f will then send D
∗
∗
to either D or D . To see which it is, choose any point z 0 in D and see whether f (z 0 ) is in D ∗
∗ ∗ ∗ ∗ ∗
or D .If f (z 0 ) is in D , then f : D → D (Figure 23.19). If f (z 0 ) is in D , then f : D → D
(Figure 23.20). In the first case, we have our conformal mapping. In the second, we do not, but
in some cases, it is possible to take another step from f and construct a conformal mapping of
D → D .
∗
We will illustrate these ideas with some examples, starting with very simple ones and
building to more difficult problems.

D D * w
z w z

f(z )
0
D z 0

C D * C * D *
* D
f : D D
C
C *
FIGURE 23.18 Domains and complementary
∗
∗
FIGURE 23.19 f : D → D if f (x 0 ) is in D .
domains.
f : D D *
D *

f(z ) w
0
z

z 0

D *
D
C
C *
∗
∗
FIGURE 23.20 f : D → D if f (z 0 ) is in D .

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 15:39 THM/NEIL Page-766 27410_23_ch23_p751-788