• Support PF! Buy your school textbooks, materials and every day products Here!

Integral with residues

  • Thread starter nicksauce
  • Start date
nicksauce
Science Advisor
Homework Helper
1,272
5
1. Homework Statement
Evaluate [tex]\int_0^{\infty} \frac{dx}{1+x^{100}}[/tex]


2. 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.


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?
 

Answers and Replies

Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,843
17
At the very least, have you written down what the summation would be?
 
nicksauce
Science Advisor
Homework Helper
1,272
5
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?
 
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,843
17
nicksauce
Science Advisor
Homework Helper
1,272
5
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?
 
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,843
17
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?
 

Related Threads for: Integral with residues

Replies
1
Views
1K
  • Last Post
Replies
6
Views
530
Replies
1
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
7
Views
1K
Replies
2
Views
2K
Top