1738 CHAPTER 55. COMPLEX MAPPINGS

which shows that the orientation of γ ′ (t0), η ′ (s0) is the same as the orientation of

( f ◦ γ)′ (t0) ,( f ◦η)′ (s0)

Mappings which preserve both orientation and angles are called conformal mappings andthis has shown that analytic functions are conformal mappings if the derivative does notvanish.

55.2 Fractional Linear Transformations55.2.1 Circles And LinesThese mappings map lines and circles to either lines or circles.

Definition 55.2.1 A fractional linear transformation is a function of the form

f (z) =az+bcz+d

(55.2.1)

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 55.2.2 The fractional linear transformation, 55.2.1 can be written as a finite com-position of dilations, inversions, and translations.

Proof: Let

S1 (z) = z+dc,S2 (z) =

1z,S3 (z) =

(bc−ad)c2 z

andS4 (z) = z+

ac

in the case where c ̸= 0. Then f (z) given in 55.2.1 is of the form

f (z) = S4 ◦S3 ◦S2 ◦S1.

Here is why.

S2 (S1 (z)) = S2

(z+

dc

)≡ 1

z+ dc

=c

zc+d.

Now consider

S3

(c

zc+d

)≡ (bc−ad)

c2

(c

zc+d

)=

bc−adc(zc+d)

.