Answer to Question #173748 in Complex Analysis for Rayhan

Question

A fixed point of f(z) is a complex number z such that f(z)=z. How many fixed points does f(z) have?

 

Answer 1

We consider the function, $g(z)=f(z)-z.$ Then the problem reduces to how many zeros does $g$ have. Now the number of zeros of an analytic function is discrete and hence countable. So question rises whether it is possible to have a function with finite or countable number of zeros. It is possible. Like $g(z)= e^{z}$ has no zeros. Any polynomial of degree $n$ has $n$ zeros, and functions like $g(z)=sin \ z$ has infinite but countable zeros. So number of zeros or equivalently fixed points are 0 or any finite number or countable.