Mapping that preserves the number of poles and zeros
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 ?
complex-analysis functional-analysis
For (1): Not unless you assume something more about $g$. Let $g$ be a function such that $g(z) = z$ in a neighbourhood $U$ of the set $S$ of poles and zeros of $f$, and $g(z)$ maps the complement of $U$ in an arbitrary way to a set on which $|f|$ is bounded above and below. Then $f \circ g$ (which is not analytic except in $U \backslash S$, but you didn't say it had to be) has the same poles and zeros as $f$, but $g$ is not in general conformal.
For (2) and (3): Linear transformations always work, and there are infinitely many of them.
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