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Evaluation of definite integral

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Homework Statement

Need to evaluate int(-inf,inf)(x/(x^3+1)).



Homework Equations





The Attempt at a Solution

I don't believe finding the residues will be a problem. However, the integral is over the whole real line, and there is a singularity at -1 on the real line, so I'm not sure how to draw an appropriate contour to integrate around. Am I supposed to integrate over a keyhole contour avoiding the singularity at -1 from the right and then from the left? Any help / advice will be much appreciated.
 
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Answers and Replies

  • #2
gabbagabbahey
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You might want to check if the integral actually exists/converges before worrying about contours.
 
  • #3
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You might want to check if the integral actually exists/converges before worrying about contours.
My book says it is equal to pi/sqrt(3) :)
 
  • #4
gabbagabbahey
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My book says it is equal to pi/sqrt(3) :)
Hmm...

[tex]\lim_{\epsilon\to 0} \left[\int_{-\infty}^{-1-\epsilon} \frac{x}{x^3+1}dx+\int^{\infty}_{-1+\epsilon} \frac{x}{x^3+1}dx\right]=\frac{\pi}{\sqrt{3}}[/tex]

But, I don't think you can say that is the same thing as [itex]\int_{-\infty}^{\infty} \frac{x}{x^3+1}dx[/itex]. Mathematica seems to agree with me, that the integral doesn't converge. I don't think [itex]x=-1[/itex] is a removable singularity.
 
  • #5
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Yeah... MATLAB seems to say the same thing.
 

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