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Residues homework help

  • Thread starter jdz86
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  • #1
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Homework Statement



Use residues to evaluate the improper integral:

[tex]\int^{\infty}_{- \infty}[/tex] [tex]\frac{cos(x)dx}{(x^{2} + a^{2})(x^{2} + b^{2})}[/tex] = [tex]\frac{\pi}{a^{2} - b^{2}}[/tex] ( [tex]\frac{e^{-b}}{b}[/tex] - [tex]\frac{e^{-a}}{a}[/tex] )

Homework Equations



a>b>0

The Attempt at a Solution



If someone could just explain the concept of residues and how to apply them, that would be great. They give the answer, but you have to use residues to show how to get there. I've looked over my notes and through the book and I don't understand them at all.
 

Answers and Replies

  • #2
HallsofIvy
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The integrand, [itex]cos(x)/(x^2+ a^2)(x^2+ b^2)[/itex] is analytic everywhere except at [itex]x=\pm ai[/itex] and [itex]x= \pm bi[/itex]. If you take a contour integral where the contour is the x-axis from (-R, 0) to (R,0) together with the half circle from (R,0) to (R, 0), you should be able to show that the integral over the half circle is 0 so that the integral over that contour is just the integral from [itex]-\infty[/itex] to [itex]\infty[/itex] as you want. Of course, that is equal to [itex]1/(2\pi i) times the sum of the residues at x= ai and x= bi which you should be able to caculate.
 

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