402 CHAPTER 10 • CONFORMAL MAPPING
11. If f is analytic at Zo and/' (zo) # 0, show that the function g (z) = f (z) preserves
the magnitude, but reverses the sense, of angles at zo.
12. If w = f (z) is a mapping, where f (z} is not analytic, then what behavior would
you expect regarding the angles between curves?10.2 Bilinear Transformations
Another important class of elementary mappings was studied by Augustus Fer-
dinand Mobius (1790-1868). These mappings are conveniently expressed as the
quotient of two linear expressions. They arise naturally in mapping problems
involving the function Arctanz. In this section, we show how they are used to
map a disk one-to-one and onto a half-plane.
If we let a, b, c, and d denote four complex constants with the restriction
that ad # be, then the functionw=S(z)=az+ b
cz+d(10-13)is called a bilinear trans formation, a Mo b ius transformation, or a linear
fractional t ransformation. If the expression for S in Equation (10-13) is
multiplied by the quantity cz + d, then the resulting expression has the bilinear
form cwz -az + dw -b = O. We collect, terms involving z and write z (cw - a) =- dw + b. Then, for values of w # ~, the inverse transformation is given by
8_ 1 ( ) -dw+b
z= w =.
cw-a(10-14)We can extend Sand s-^1 to mappings in the extended complex plane. The
value S (oo) should equal the limit of S (z) as z-+ oo. T herefore, we definea+ (k) a
S(oo) = lim S(z) = lim (;) = - ,
z~oo .z~oo c + z Cand the inverse is s-^1 (~) = oo. Similarly, t he value s-^1 (oo) is obtained by
-d+ (11.) -d
s-^1 (oo) = Jim s-^1 (w) = lim w = -,
w~oo w-oo c - (~) Cand the inverse is S (-;,d) = oo. With these extensions we conclude that the
transformation w = S (z) is a one-to-one mapping of the extended complex z
plane onto the extended complex w plane.
·we now show that a bilinear transformation carries the class of circles and
lines onto itself. If S is an arbitrary bilinear transformation given by Equation