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Integral of 1/sqrt(x)exp(-ix) dx

  1. Mar 26, 2013 #1

    VVS

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    Hi,

    I am just doing this out of curiousity.

    1. The problem statement, all variables and given/known data

    I want to integrate 1/sqrt(x)exp(ix) dx from minus infinity to infinity.


    2. Relevant equations



    3. The attempt at a solution

    I had a couple of ideas one was to substitute x=u^2
    but then you mess up the limits and you get minus imaginary infinity.

    The other idea was to use Jordan's Lemma. But as far as I know 1/sqrt(x) doesn't have a residue so it can't be applied.

    How do you solve this integral then?

    thank you
     
  2. jcsd
  3. Mar 26, 2013 #2

    mfb

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    2016 Award

    Staff: Mentor

    That substitution looks like a good idea, if you split up your integral into two parts first.
     
  4. Mar 26, 2013 #3

    VVS

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    Hey,

    Thanks for the hint. I have done the splitting, but I am not sure about the limits.

    If I set u=sqrt(x) then the lower limit is plus or minus imaginary infinity and moreover the upper limit can take two values: plus infinity or minus infinity.

    I don't know how to integrate that. I am aware that cos(x^2)dx and sin(x^2)dx are the well known fresnel integrals.

    thanks for your help.
     
  5. Mar 26, 2013 #4

    mfb

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    Staff: Mentor

    For x from 0 to plus (real) infinity, u goes from 0 to plus (real) infinity.
    For negative x, you can transform the integral to the integral for positive x.
     
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