How does one show that the residue of the first derivative of a holomorphic function is zero in its singularity?

Question

I've got a holomorphic function $f:\mathbb{C}\backslash\{0\}\rightarrow\mathbb{C}$ and I want to show that $$res_{0}f'=0$$

Answer 1

Applying the definition of Residue as the -1'th Laurent series coefficient: $$\operatorname{Res}(f';0)=a_{-1}=\frac{1}{2\pi i} \oint_{\gamma}\,\frac{f'(z)}{(z-c)^{-1+1}}\,dz = \frac{1}{2\pi i} \oint_{\gamma}\,f'(z)\,dz .$$

And $f'$ has a primitive along all of $\gamma$ so this integral is $0$.