I am sorry if the following questions have been answered before...
Suppose you have a function, $f(z)$ of a complex variable, $z$. The function $f(z)$ has a $m$ number of poles and $n$ number of zeros in the complex plane.
1) Suppose there is a mapping $g$, that acts on $z$ such that $f(g(z))$ has the same number of poles and zeros as $f(z)$. Is the map $g$ a subclass of what is known as a ''Conformal map'' ? Or do such functions go by any special name ?
2) For a given $f(z)$, can I always find a mapping $g$ that preserves the number of poles and zeros in the complex plane?
3) Also, given that such maps exist for $f(z)$, is the number of distinct $g(z)$ s usually infinite ? Can they form a finite set ?
Thanks in advance.