Just a quick True-False question from Complex Variables - Francis J. Flanigan
Could someone confirm my reasoning?
True or false?
If $f(z)$ holomorphic at the origin, then there exists an integer $n$ such that $g(z) = \dfrac{f(z)}{z^m}$ has a pole at the origin, provided $m>n$.
I would argue false since $f(z) = 0$ does not meet the conditions, while the book states 'true'.
I think if the problem would state "... nonconstant function $f(z)$ ..." then $f(z)$ could be written as $f(z) = (z-z_0)^n g(z)$ where $g(z_0)\not = 0$ and this would meet the conditions.
You are correct. For every $m\in \Bbb{N}$, the function $z\mapsto \frac{0}{z^m}$ has a removable singularity at the origin.