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Homework Help: Integral Trouble

  1. Sep 5, 2005 #1
    While doing some quantum mechanics homework I came accross 3 integrals that are giving me trouble.


    1) dx/(a^2+x^2)^2 from negative infinity to infinity

    2) (x^2 dx)/(a^2+x^2)^2 from negative infinity to infinity

    3) (sinkx)^2/x^2 from 0 to infinity

    I would appreciate any help that anyone has to offer. #3 resembles the Dirichlet Integral (if we drop off the squares) and then it would be pi/2. All k's and a's are just constants.

    Thanks,

    jcain6
     
  2. jcsd
  3. Sep 6, 2005 #2
    For the first one, try [itex]x=a\tan{\theta}[/itex]. That will be really messy though, there must be a better way...
     
  4. Sep 6, 2005 #3

    lurflurf

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    For (2) use "integration by parts"
    [tex]\int_a^b u \ dv=uv|_a^b-\int_a^b v \ du[/tex]
    chose u=x dv=x/(x^2+a^2)^2
    for (1) note
    a^2(1)+(2)=pi/a
    for (1) and (2) I assume a>0
    for (3)
    differentiate w/ respect k to obtain a Dirichlet Integral
    note (3)=0 if k=0 then integrate w/respect k to find (3)
     
    Last edited: Sep 7, 2005
  5. Sep 6, 2005 #4
    I suggest you try to use some complex analysis. It seemed to work for me, on the first one anyway.
     
  6. Sep 6, 2005 #5
    I'm curious. How did you do this?
     
  7. Sep 7, 2005 #6

    lurflurf

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    Residue Theorem
    [tex]\oint_C f(z) \ dz=2\pi i\sum res(f(z))_{z=a}[/tex]
    In words the Integral of a function around a closed contour equals the sum of the residues of the sigularities of the function times 2pi*i. For (1) and (2) the functions themselves with a contour of a large semicircle in the upper half plan works fine. The sine one is a bit tricker use f(z)=1-exp(2kiz) and a contour of semicircle in upperhalf plane with indentation at origin. There is however no need to use the residue theorem as I above noted the easy way to do them.
     
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