Integral with residues

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  • #1
nicksauce
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Homework Statement


Evaluate [tex]\int_0^{\infty} \frac{dx}{1+x^{100}}[/tex]


Homework 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.


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?
 

Answers and Replies

  • #2
Hurkyl
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At the very least, have you written down what the summation would be?
 
  • #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?
 
  • #4
Hurkyl
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Does that look okay so far?
I expected something like that; why'd you stop there?
 
  • #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?
 
  • #6
Hurkyl
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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?
Yes; this is a kind of sequence you're quite familiar with. How are consecutive terms related?
 

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