Complex Analysis - Maths KU

430 Chapter 7 Conformal Mappings
y
∇ = 0
2
Φ
v
∇ φ = 0
2
φ = 30 φ = 0
1
(a) Dirichlet problem
Φ=
0
r 0
2 3
Φ = 30
(b) Transformed Dirichlet problem
Figure 7.48 Figure for Example 1
1
x
u
In the following examples, we will applysome ideas from the preceding
sections in this chapter to help solve Dirichlet problems arising in the areas
of electrostatics, fluid flow, and heat flow. Recall from Section 3.3 that if
a function φ(x, y) satisfies Laplace’s equation (1) in some domain D, then
φ(x, y) is harmonic in D. Moreover, if ψ(x, y) is a harmonic conjugate of
φ(x, y) inD, then the function
Ω(z) =φ(x, y)+iψ(x, y)
is analytic in D and is called a complex potential function. The level
curves of φ and ψ have important physical interpretations in applied mathematics.
Their interpretations are summarized in Table 3.1 of Section 3.4.
EXAMPLE 1 A Heat Flow Application
Determine the steady-state temperature φ in the domain D consisting of all
points outside of the two circles |z| = 1 and � �
� 5 z − � 1
2 = 2 , shown in color in
Figure 7.48(a), that satisfies the indicated boundaryconditions.
Solution The steady-state temperature φ is a solution of Laplace’s equation
(1) in D that satisfies the boundaryconditions
φ(x, y) = 30 if x 2 + y 2 =1,
φ(x, y) =0 if � x − 5
�2 2 1
2 + y = 4 .
We solve this problem using the four steps given on page 429.
Step 1 EntryC-1 in Appendix III indicates that we can map D onto an
annulus. Identifying b = 2 and c = 3 in EntryC-1, we find that
and
a = bc +1+� (b 2 − 1) (c 2 − 1)
b + c
r0 = bc − 1 − � (b 2 − 1) (c 2 − 1)
c − b
= 7+2√6 ,
5
=5− 2 √ 6.
Thus, the domain D is mapped onto the annulus 5 − 2 √ 6