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Integral with residues

  1. Apr 26, 2008 #1

    nicksauce

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    1. The problem statement, all variables and given/known data
    Evaluate [tex]\int_0^{\infty} \frac{dx}{1+x^{100}}[/tex]


    2. Relevant equations
    [tex]\int^{\infty}_{-\infty} \frac{P(x)dx}{Q(x)} = 2\pi i\sum_{\textnormal{res}}\frac{P(z)}{Q(z)}[/tex] in the Upper half plane.


    3. The attempt at a solution
    I really can't be expected to calculate the residue of this function some 50 times can I? There must be some trick I am missing. Any hints?
     
  2. jcsd
  3. Apr 26, 2008 #2

    Hurkyl

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    At the very least, have you written down what the summation would be?
     
  4. Apr 26, 2008 #3

    nicksauce

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    Ok so I need to sum all the residues of
    [tex]\frac{1}{1+z^{100}}[/tex] in the upper half plane. There are simple poles at [tex]z=e^{i\pi/100},e^{3i\pi/100},e^{5i\pi/100}...e^{99i\pi/100}[/tex].

    The residue at the first pole is
    [tex]\lim_{z\rightarrow e^{i\pi/100}} \frac{z-e^{i\pi/100}}{1+z^{100}}[/tex]
    [tex] = \frac{1}{100e^{99i\pi/100}}[/tex]

    The residue at the nth pole (the first being the zeroth) will be:
    [tex] \frac{1}{100e^{(1+2n)i\pi*99/100}}[/tex]

    Does that look okay so far?
     
  5. Apr 26, 2008 #4

    Hurkyl

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    I expected something like that; why'd you stop there?
     
  6. Apr 26, 2008 #5

    nicksauce

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    Okay so then I can get

    Sum of residues =
    [tex]\frac{1}{100}\sum_{n=0}^{n=49}e^{-(1+2n)i\pi99/100}[/tex]

    Any way to do this analytically?
     
  7. Apr 26, 2008 #6

    Hurkyl

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    Yes; this is a kind of sequence you're quite familiar with. How are consecutive terms related?
     
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