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Complex integral

  1. Nov 21, 2013 #1
    1. The problem statement, all variables and given/known data
    I have an integral of the form:

    0exp(ax+ibx)/x dx
    What is the general method for solving an integral of this kind.

    2. Relevant equations
    Maybe residual calculus?


    3. The attempt at a solution
     
  2. jcsd
  3. Nov 21, 2013 #2
    Residues sound like the only way. However, make sure you are using ##e^{-z}/z##; with a positive exponent your integral will not converge.
     
  4. Nov 21, 2013 #3

    Dick

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    I don't see how it's going to converge no matter what a is. It's also divergent near x=0.
     
  5. Nov 21, 2013 #4
    So true. Wish I had read the lower limit as 0 instead of 1. Can I blame it on bad eyesight?
     
  6. Nov 22, 2013 #5
    I need it to converge badly. But I know what mistake I made. I wanted the integral to be the imaginary part of the above. At least I think so. I have attached the whole exercise now as pdf. Is it correct what I have done so far and how do I evaluate the integral?
     

    Attached Files:

    • FT.pdf
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    Last edited: Nov 22, 2013
  7. Nov 22, 2013 #6

    Ray Vickson

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    It all looks incorrect. You wanted to integrate something like ##\exp(-cr)/r \, \exp(ikr\ cos(\theta))## over ##R^3## in spherical corrdinates. The volume element in spherical coordinates is not ##dr \, d\theta \, d\phi##; it is ##r^2 \sin(\theta)\, dr \, d \theta \, d \phi##.
     
  8. Nov 23, 2013 #7
    Right, I wrote that in a rush I can see. So basically I forgot the sin(theta) in the first line but it should be there or I couldnt make the substitution dcos(theta). Also the r^2 should be there and 1/r I forgot too so I would end up with having to integrate the imaginary part of r times the expression on the last line. But still with that, I don't see how I can solve that integral.
     
  9. Nov 23, 2013 #8

    Ray Vickson

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    Your final integral is "elementary" and is the type of thing you learned to do in Calculus 101. Look at it again.
     
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