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Quick question on Laurent series proof uniqueness

  • Thread starter binbagsss
  • Start date
  • #1
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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
 

Answers and Replies

  • #2
stevendaryl
Staff Emeritus
Science Advisor
Insights Author
8,401
2,577
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}##
 

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