Quick question on Laurent series proof uniqueness

[binbagsss]

Homework Statement

I am looking at the wikipedia proof of uniqueness of laurent series:

https://en.wikipedia.org/wiki/Laurent_series

Homework Equations

look above or below

The Attempt at a Solution

I just don't know what the indentity used before the bottom line is, I've never seen it before, would someone kindlly explain this to me or point me to a link?

Many thanks

 

[stevendaryl]

Let's integrate the function $1/z^n$ in the complex plane around a loop enclosing the origin, $z=0$. In this loop, let $z = R e^{i\theta}$ so $dz = iR e^{i\theta} d\theta$. So ($\theta$ goes from 0 to $2\pi$):

$\int \frac{1}{z^n} dz = \int \frac{1}{R^n e^{i n \theta}} iR e^{i\theta} d\theta$
$= \frac{i}{R^{n-1}} \int e^{(1-n) i \theta} d\theta$

If $n$ is an integer and $n \neq 1$, then we have

$\int \frac{1}{z^n} dz = \frac{1}{R^{n-1}} \frac{1}{1-n} [e^{2\pi (1-n) i} - 1] = 0$ (since $e^{2(1-n)\pi i} = 1$)

If $n=1$, then

$\int \frac{1}{z^n} dz = \int i d\theta = 2\pi i$

So we can summarize this:

$\int \frac{1}{z^n} dz = 2\pi i \delta_{n-1}$