406 CHAPTER 16. MAPPING THEOREMS

at α. Then f has the form given in 16.1. Therefore,

f ′ (z)f (z)

=h′ (z)−∑

pk=1

kbk(z−α)k+1

h(z)+∑pk=1

bk(z−α)k

=−p bp

(z−α) + s(z)

bp + r (z)

where limz→α s(z) = limz→α r (z) = 0. Thus

limz→α

(z−α)f ′ (z)f (z)

=−p = res(

f ′

f,α

),

where p is the multiplicity of the pole.Next suppose f has a zero of multiplicity p at α. Then

limz→α

(z−α)f ′ (z)f (z)

= limz→α

∑∞k=p akk (z−α)k

∑∞k=p ak (z−α)k = lim

z→α

∑∞k=p akk (z−α)k−p

∑∞k=p ak (z−α)k−p = p

and from this, res( f ′/ f ) = p, the multiplicity of the zero. The conclusion of this theoremnow follows from the residue theorem, Theorem 15.6.2. ■

16.4 Fractional Linear TransformationsThese mappings map lines and circles to either lines or circles.

Definition 16.4.1 A fractional linear transformation is a function of the form

f (z) =az+bcz+d

(16.2)

where ad−bc ̸= 0.

Note that if c = 0, this reduces to a linear transformation (a/d)z+(b/d) . Special casesof these are defined as follows.

dilations: z→ δ z, δ ̸= 0, inversions: z→ 1z,

translations: z→ z+ρ.

The next lemma is the key to understanding fractional linear transformations.

Lemma 16.4.2 The fractional linear transformation, 16.2 can be written as a finitecomposition of dilations, inversions, and translations.

Proof: If d = 0 then c ̸= 0 and 16.2 reduces to ac +

bc

( 1z

)which is recovered as

z→ 1z→ b

c

(1z

)→ b

c

(1z

)+

ac

So assume d ̸= 0. Then, using the special transformations, consider

z → 1z→ d

z→ d

z+ c =

cz+dz→ z

cz+d

→ αzcz+d

→ αzcz+d

+ p =(α + pc)z+d p

cz+d