1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers

D

I I

I I

0 A

Fig. 1.9

Q I I \ I \ \ p

39

A geometric construction of the inverse of a given point with respect to a

circle can be made in several ways. In Fig. 1.10 the inverse P^11 of P' has

been obtained by constructing first the tangent P'T to the unit circle, and

then the perpendicular T P^11 to (^0) --I P'. From --II the --2 similarity of the triangles

OP'T and OTP" it follows that OP ·OP =OT = 1, which establishes

P' and P^11 as inverse points. When P' lies inside the unit circle the steps

in the foregoing construction are reversed, and when P' lies on the circle,

P' is its own inverse.

Since 1/ Z1 = zif Z1Z1 = zi/lz11^2 shows that 1/ Z1 is in the direction of

z 1 and has a length 1/lz 1 1, we may perform the construction of l/z1 as

in Fig. 1.11.

y

Fig. 1.10

\ x

' \ \

\

' \ I

\ I

\1

'P'(Z1)