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Homework Help: Quick question on Laurent series proof uniqueness

  1. May 18, 2018 #1
    1. The problem statement, all variables and given/known data

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

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


    2. Relevant equations

    look above or below


    3. 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
     
  2. jcsd
  3. May 18, 2018 #2

    stevendaryl

    User Avatar
    Staff Emeritus
    Science Advisor

    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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