Cauchy Integral formula
Let $G$ be open in $\mathbb{C}$ and $f$ be holomorphic on $G$.
Let $\gamma$ be a closed rectifiable curve in $G$ such that $Wnd(\gamma,z)=0$ for all $z\in \mathbb{C}\setminus G$.
Then, $Wnd(\gamma,z) f(z)= \frac{1}{2\pi i}\int_\gamma \frac{f(w)}{w-z} dw$
I'm now reviewing complex analysis I have learned.
Not only Cauchy integral formula, but all theorems relating line integral comes with the coefficient $\frac{1}{2\pi i}$ in basic complex analysis.
I completely understand the proof for Cauchy integration formula and other theorems (such as Counting zeros, Residue theorem, Argument principle and etc) and I know how $2\pi i$ is derived.
However, I cannot find why $2\pi i$ must be there geometrically or by any means. All I can say for now is that $2\pi i$ incidentally came out in proofs. I'm sure it has some meaning.. What is it?
Thank you in advance.