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How to deal with this sum complex analysis?

  1. Jan 12, 2015 #1
    1. The problem statement, all variables and given/known data

    ceme4.png

    2. Relevant equations
    Down

    3. The attempt at a solution

    As you see in the solution, I am confused as to why the sum of residues is required.

    My question is the sum:

    $$(4)\cdot\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n^3}$$

    Question #1:

    -Why is the beginning n=1 the residue is for all $$n \in (-\infty, \infty)$$ so the beginning of the sum should be $$n=-\infty$$ right?

    Question #2:

    Also in the residue theorem, it states:

    $$\oint_{C} f(z) dz = (2\pi i)(\sum \text{Res})$$

    Where is the 2pi(i) gone in the solution of theirs?

    Thanks!
     
  2. jcsd
  3. Jan 12, 2015 #2

    Dick

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    Homework Helper

    The limit of the contour integral is 0 as n->infinity. So the sum of all of the pole residues is 0. So ##2 \pi i## is a common factor of all of the residues. It cancels. The residue at 0 is the ##-7 \pi^3/45## part. For n>0 the residue is the same at n, -n, ni and -ni. So you can sum them all together by adding a factor of 4. The means you only have to sum over n>0.
     
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