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23.2 Construction of Conformal Mappings 773

As an example, 0 is in D, and
√
2
iπ/4
f (0) = e = (1 + i)
2
is in the rotated half-plane u + v> 0.
23.2.1 The Schwarz-Christoffel Transformation
The Schwarz-Christoffel transformation is designed to produce a conformal mappings of the
upper half-plane U in the z-plane to the region P bounded by a polygon P in the w-plane. This
polygon could be a triangle, rectangle, pentagon, or other polyhedron. Because of the corners on
the boundary polygon P of P, such a mapping will be unlike anything we have seen up to this
point.
Let P be an n-sided polygon with vertices w 1 ,w 2 ,··· ,w n in the w-plane (Figure 23.29)
with exterior angles α 1 π,α 2 π,··· ,α n π.
The Schwarz-Christoffel conformal mapping f of the upper half-plane to the interior of P
has the form
z
f (z) = a (ξ − x 1 ) −α 1 (ξ − x 2 ) −α 2 ···(ξ − x n ) −α n dξ + b (23.3)
z 0
in which x 1 , x 2 ,··· , x n are real numbers labeled in increasing order, a and b are complex num-
bers, and z 0 is a complex number with Im(z 0 )> 0. These numbers must be chosen to suit P.The
integral is taken over any path in U from z 0 to z in U. The factors (ξ − x j ) −α j are defined using
the complex logarithm obtained by restricting the argument to [0,2π).
To dissect this expression for f (z) and see the ideas behind the various components, let
g(z) = a(ξ − x 1 ) −α 1 (ξ − x 2 ) −α 2 ···(ξ − x n ) −α n
for z in U. Then f (z) = g(z), and

arg( f (z)) = arg(a) − α 1 arg(z − x 1 ) − ··· − α n arg(z − x n ).
As we saw in the proof of Theorem 23.1, arg( f (z)) is the number of radians by which f (as a

mapping) rotates tangent lines if f (z) = 0.

Now imagine z moving from left to right along the real axis, which is the boundary of U.On
(−∞, x 1 ) to the left of x 1 , f (z) moves along a straight line (no change in the angle). As z passes
over x 1 ,arg( f (z)) changes by α 1 π. This angle remains fixed as z moves from x 1 to x 2 .As z

w
α π
1
v w 1
α π w 2
2
w 3
α π
3

u

FIGURE 23.29 A polygon and its exterior
angles.

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October 14, 2010 15:39 THM/NEIL Page-773 27410_23_ch23_p751-788