Fixed Points of a Complex Function

A fixed point of a complex functionis somesatisfying

For example, ifthenimplies

The behaviour offor values ofclose todepends on the derivative ofatIfthenmaps small discs with centreto even smaller discs with centreThis means that values ofclose toare attracted toby iteration.

Theorem

Letbe a fixed point of an analytic functionand suppose thatThen there existssuch thatfor

Proof

Choosesuch that

Sinceandthere is a positive numbersatisfying for

Hencefor

So, since for

Thus ifthen

Hence

In generalbutis a null sequence so

Not all fixed points exhibit this behaviour.

A fixed pointof an analytic complex functionis

a) attracting if

b) repelling if

c) indifferent if

d) super – attracting if