Complex Analysis - Maths KU

416 Chapter 7 Conformal Mappings
formula (6) with n − 1 = 2 of the interior angles. After selecting x1 = 0 and
x2 = 1, (6) gives
f ′ (z) =Az −2/3 (z − 1) −2/3 . (12)
There is no antiderivative of the function in (12) that can be expressed in
terms of elementaryfunctions. However, f ′ is analytic in the simply connected
domain y>0, and so, from Theorem 5.8 of Section 5.4, an antiderivative f
does exist in this domain. The antiderivative is given bythe integral formula
� z
f(z) =A
0
1
s2/3 ds + B, (13)
2/3
(s − 1)
where A and B are complex constants. Requiring that f(0) = 0 allows us to
solve for the constant B. Since � 0
0 =0,wehave
� 0
f(0) = A
0
1
s2/3 ds + B =0+B = B,
2/3
(s − 1)
and so f(0) = 0 implies that B = 0. If we also require that f(1) = 1, then
� 1
f(1) = A
0
Let Γ denote value of the integral
Γ=
� 1
Then A =1/Γ and f can be written as
0
f(z) = 1
� z
Γ 0
1
s2/3 ds =1.
2/3
(s − 1)
1
s2/3 ds.
2/3
(s − 1)
1
s2/3 ds.
2/3
(s − 1)
Values of f can be approximated using a CAS. For example, using the
NIntegrate command in Mathematica we find that
f(i) ≈ 0.4244 + 0.3323i and f(1 + i) ≈ 0.5756 + 0.3323i.
The Schwarz-Christoffel formula can also sometimes be used to find mappings
onto nonpolygonal regions. Such mappings are often needed in the
studyof ideal fluid flows. The Schwarz-Christoffel formula can be used when
the desired nonpolygonal region can be obtained as a “limit” of a sequence of
polygonal regions. The following example illustrates this technique.