What is a geometric meaning of $2\pi i$ for complex integration?

Question

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.

Answer 1

Definition. Let $\gamma:[0,1]\rightarrow\mathbb{C}$ be a loop and let $a\in\mathbb{C}\setminus\gamma([0,1])$. Let define: $$\textrm{Wnd}(\gamma,a):=\frac{1}{2i\pi}\int_{\gamma}\frac{\mathrm{d}z}{z-a}.$$ $\textrm{Wnd}(\gamma,a)$ is the winding number of $\gamma$ around $a$.

One has the:

Proposition. Let $\gamma:[0,1]\rightarrow\mathbb{C}$ be a loop and let $a\in\mathbb{C}\setminus\gamma([0,1])$, $\textrm{Wnd}(\gamma,a)$ is an integer.

Without the factor $\displaystyle\frac{1}{2i\pi}$, $\textrm{Wnd}(\gamma,a)$ will not be an integer and one cannot have the following:

Geometric interpretation. $\textrm{Wnd}(\gamma,a)$ is the algebraic number of turns of $\gamma$ around the point $a$, that is the number of trigonometric turns of $\gamma$ around $a$ minus the number of clockwise turns of $\gamma$ around $a$.

Example. To get insight let us do the following computation. Let $n\in\mathbb{Z}$ and let define the loop $\gamma_n:[0,1]\ni t\mapsto e^{2in\pi t}\in\mathbb{S}^1$, one has: $$\textrm{Wnd}(\gamma_n,0)=\frac{1}{2i\pi}\int_{0}^1\frac{\gamma'(t)}{\gamma(t)}\mathrm{d}t=\frac{1}{2i\pi}\int_0^1\frac{2in\pi e^{2in\pi t}}{e^{2in\pi t}}\mathrm{d}t=n.$$ $\gamma_n$ make $|n|$ turns around the origin, those turns are trigonometric turns if $n\geqslant0$ and clockwise turns if $n<0$.

Answer 2

$$i 2 \pi = \oint_{\gamma} \frac{dz}{z-w} $$

when $w$ is inside $\gamma$ and $\gamma$ only winds around $w$ once counterclockwise (positive orientation).