55.2. FRACTIONAL LINEAR TRANSFORMATIONS 1739

Finally, consider

S4

(bc−ad

c(zc+d)

)≡ bc−ad

c(zc+d)+

ac=

b+azzc+d

.

In case that c = 0, f (z) = ad z+ b

d which is a translation composed with a dilation. Becauseof the assumption that ad− bc ̸= 0, it follows that since c = 0, both a and d ̸= 0. Thisproves the lemma.

This lemma implies the following corollary.

Corollary 55.2.3 Fractional linear transformations map circles and lines to circles orlines.

Proof: It is obvious that dilations and translations map circles to circles and lines tolines. What of inversions? If inversions have this property, the above lemma implies ageneral fractional linear transformation has this property as well.

Note that all circles and lines may be put in the form

α(x2 + y2)−2ax−2by = r2−

(a2 +b2)

where α = 1 gives a circle centered at (a,b) with radius r and α = 0 gives a line. In termsof complex variables you may therefore consider all possible circles and lines in the form

αzz+β z+β z+ γ = 0, (55.2.2)

To see this let β = β 1 + iβ 2 where β 1 ≡ −a and β 2 ≡ b. Note that even if α is not 0 or 1the expression still corresponds to either a circle or a line because you can divide by α ifα ̸= 0. Now I verify that replacing z with 1

z results in an expression of the form in 55.2.2.Thus, let w = 1

z where z satisfies 55.2.2. Then(α +βw+βw+ γww

)=

1zz

(αzz+β z+β z+ γ

)= 0

and so w also satisfies a relation like 55.2.2. One simply switches α with γ and β with β .Note the situation is slightly different than with dilations and translations. In the case of aninversion, a circle becomes either a line or a circle and similarly, a line becomes either acircle or a line. This proves the corollary.

The next example is quite important.

Example 55.2.4 Consider the fractional linear transformation, w = z−iz+i .

First consider what this mapping does to the points of the form z = x+ i0. Substitutinginto the expression for w,

w =x− ix+ i

=x2−1−2xi

x2 +1,

a point on the unit circle. Thus this transformation maps the real axis to the unit circle.The upper half plane is composed of points of the form x+ iy where y > 0. Substituting

in to the transformation,

w =x+ i(y−1)x+ i(y+1)

,