1740 CHAPTER 55. COMPLEX MAPPINGS

which is seen to be a point on the interior of the unit disk because |y−1| < |y+1| whichimplies |x+ i(y+1)| > |x+ i(y−1)|. Therefore, this transformation maps the upper halfplane to the interior of the unit disk.

One might wonder whether the mapping is one to one and onto. The mapping is clearlyone to one because it has an inverse, z = −i w+1

w−1 for all w in the interior of the unit disk.Also, a short computation verifies that z so defined is in the upper half plane. There-fore, this transformation maps {z ∈ C such that Imz > 0} one to one and onto the unit disk{z ∈ C such that |z|< 1} .

A fancy way to do part of this is to use Theorem 52.3.5. limsupz→a∣∣ z−i

z+i

∣∣≤ 1 whenevera is the real axis or ∞. Therefore,

∣∣ z−iz+i

∣∣≤ 1. This is a little shorter.

55.2.2 Three Points To Three PointsThere is a simple procedure for determining fractional linear transformations which map agiven set of three points to another set of three points. The problem is as follows: There arethree distinct points in the extended complex plane, z1,z2, and z3 and it is desired to finda fractional linear transformation such that zi → wi for i = 1,2,3 where here w1,w2, andw3 are three distinct points in the extended complex plane. Then the procedure says that tofind the desired fractional linear transformation solve the following equation for w.

w−w1

w−w3· w2−w3

w2−w1=

z− z1

z− z3· z2− z3

z2− z1

The result will be a fractional linear transformation with the desired properties. If any ofthe points equals ∞, then the quotient containing this point should be adjusted.

Why should this procedure work? Here is a heuristic argument to indicate why youwould expect this to happen rather than a rigorous proof. The reader may want to tightenthe argument to give a proof. First suppose z = z1. Then the right side equals zero and sothe left side also must equal zero. However, this requires w = w1. Next suppose z = z2.Then the right side equals 1. To get a 1 on the left, you need w = w2. Finally supposez = z3. Then the right side involves division by 0. To get the same bad behavior, on the left,you need w = w3.

Example 55.2.5 Let Imξ > 0 and consider the fractional linear transformation whichtakes ξ to 0, ξ to ∞ and 0 to ξ/ξ , .

The equation for w isw−0

w−(

ξ/ξ

) =z−ξ

z−0· ξ −0

ξ −ξ

After some computations,

w =z−ξ

z−ξ.

Note that this has the property that x−ξ

x−ξis always a point on the unit circle because it is a

complex number divided by its conjugate. Therefore, this fractional linear transformation