Holomorphic function and removable singularity

Question

If a function is holomorphic on $\Bbb D$\ ${0}$, does that mean 0 is a removable singularity of the function if the residue at 0 is 0?

I think this is false because I thought I did some practice problems and sometimes the residue at poles can be 0 too. But can someone please offer me a proof? Thanks.

Answer 1

All kinds of singularities are possible:

$f(z)=\frac{\sin z}{z}$ has a removable singularity at $0$

$f(z)=\frac{1}{z^{123456}}$ has a pole at $0$

$f(z)=e^{1/z}$ has an essential singularity at $0$