Contour integration

  • Thread starter xdrgnh
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  • #1
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Homework Statement



Integrate [e^(bx)]/(1+e^x)dx from inf to -inf

0<b<1

Homework Equations



Sum of the residues times 2pi equals the integral with a path that encloses the residues. That path includes the complex part of path which is above the x axis and the real part which is a straight line on the x axis. It's a semi circle.


The Attempt at a Solution



So I put this into the complex plain and at (pi)i as one of poles and calculate the residue there. However usually the complex curve part goes to zero but in this case it doesn't. Also I never saw an integral in this form before. I know integrals in the form (x^-a)/(1+x) and trig integrals using the Jorden Lemma. Any help will be appreciated.
 
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Answers and Replies

  • #2
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[itex] e^{bx}=(e^b)^x [/itex] and because b<1,if we'll tend to zero when x tends to infinity.Also [itex]1+e^x [/itex] tends to infinity when x tends to infinity.So you can use Jordan's lemma.
 
  • #3
vela
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I suggest you start with the substitution ##u=e^x## first.
 
  • #4
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sorry guys,what I said was wrong.b<1 doesn't mean we can use Jordan's lemma.
 
  • #5
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So I put this into the complex plain and at (pi)i as one of poles and calculate the residue there. However usually the complex curve part goes to zero but in this case it doesn't. Also I never saw an integral in this form before. I know integrals in the form (x^-a)/(1+x) and trig integrals using the Jorden Lemma. Any help will be appreciated.

How about using a rectangle enclosing just the pole at ##\pi i##?

And by the way, you mind showing how you're computing the residue(s) because even the one at ##\pi i## is not in my opinion, too easy to compute.
 
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