Mathematical Methods for Physics and Engineering : A Comprehensive Guide

24.7 CONFORMAL TRANSFORMATIONS

x 1 x 2 x 3 x 4 x 5

w 1

w 2

w 3

w 4

w 5

φ 1

φ 2 φ 3

φ 4

φ 5

y

x

s

r

w=g(z)

Figure 24.5 Transforming the upper half of thez-plane into the interior

of a polygon in thew-plane, in such a way that the pointsx 1 ,x 2 ,...,xnare

mapped onto the verticesw 1 ,w 2 ,...,wnof the polygon with interior angles

φ 1 ,φ 2 ,...,φn.

z 0 =i. By further requiringz=±∞to be mapped ontow= 1, we find 1 =w=expiφ
and soφ= 0. The required transformation is therefore

w=

z−i

z+i

,

and is illustrated in figure 24.4.

We conclude this section by mentioning the rather curiousSchwarz–Christoffel

transformation.§Suppose, as shown in figure 24.5, that we are interested in a

(finite) number of pointsx 1 ,x 2 ,...,xnon the real axis in thez-plane. Then by

means of the transformation

w=

{

A

∫z

0

(ξ−x 1 )(φ^1 /π)−^1 (ξ−x 2 )(φ^2 /π)−^1 ···(ξ−xn)(φn/π)−^1 dξ

}

+B, (24.30)

we may map the upper half of thez-plane onto the interior of a closed polygon in

thew-plane havingnverticesw 1 ,w 2 ,...,wn(which are the images ofx 1 ,x 2 ,...,xn)

with corresponding interior anglesφ 1 ,φ 2 ,...,φn, as shown in figure 24.5. The

real axis in thez-plane is transformed into the boundary of the polygon itself.

The constantsAandBare complex in general and determine the position,

size and orientation of the polygon. It is clear from (24.30) thatdw/dz=0at

x=x 1 ,x 2 ,...,xn, and so the transformation is not conformal at these points.

There are various subtleties associated with the use of the Schwarz–Christoffel

transformation. For example, if one of the points on the real axis in thez-plane

(usuallyxn) is taken at infinity, then the corresponding factor in (24.30) (i.e. the

one involvingxn) is not present. In this case, the point(s)x=±∞are considered

as one point, since they transform to a single vertex of the polygon in thew-plane.

§Strictly speaking, the use of this transformation requires an understanding of complex integrals,

which are discussed in section 24.8.