Page 783 - Advanced_Engineering_Mathematics o'neil

23.1 Conformal Mappings 763

z

N : (0, 0, 2)

S(x, y)

y

(x, y)

x

FIGURE 23.16 Stereographic
projection and the complex sphere.

point on a simple surface, and we can envision a point on the complex sphere approaching N on
the sphere more easily and concretely than we can envision z →∞ in the plane.
In defining bilinear transformations, it is sometimes convenient to map one of the three given
points in Theorem 23.3 to infinity. The next theorem and its proof tell how to do this.

THEOREM 23.4

Let z 1 , z 2 , z 3 be three distinct numbers in the z-plane, and let w 1 ,w 2 be distinct numbers in the
w-plane. Then there is a bilinear transformation T that maps
T (z 1 ) = w 1 , T (z 2 ) = w 2 , T (z 3 ) =∞.

Proof T is obtained by solving for w in the equation
(w 1 − w)(z 1 − z 2 )(z 3 − z) = (z 1 − z)(z 1 − z 2 )(w 1 − w 2 ). (23.2)
We can therefore map z 3 to infinity by deleting the factors in equation (23.1) that involve w 3 .

EXAMPLE 23.9
We will find a bilinear transformation mapping

i → 4i,1 → 3 − i,2 + i →∞.
Solve for w in the equation
(4i − w)(i − 1)(2 + i − z) = (i − z)(−3 + 5i)(1 + i)

to obtain
(5 − i)z − 1 + 3i
w = T (z) = .
−z + 2 + i
It is routine to check that T (i) = 4i, T (1) = 3 − i, and T (2 + i) =∞.

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