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Integral of 1/sqrt(x)exp(-ix) dx using integration by parts

  1. Mar 28, 2013 #1

    VVS

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    Hi,

    1. The problem statement, all variables and given/known data
    I have already evaluated the integral 1/sqrt(x)exp(-ix) using The Residue Theorem and now I was looking for another method. So I thought of applying integration by parts and I got this attached formula.

    integral.jpg

    Now I am wondering how to evaluate this series. My first doubt is whether it converges or not.
    Actually I just want to evaluate it for n=1.
    Does anyone have any ideas? Can this sum be expressed as an integral?



    Thank you
     
  2. jcsd
  3. Mar 28, 2013 #2

    haruspex

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    I don't see much hope with that series, but how about applying the 2D polar coordinates trick used to calculate the integral of exp(-x2) from 0 to ∞?
     
  4. Mar 29, 2013 #3

    VVS

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    Hey,

    Yeah I know it is way to complicated.
    But one question: If I know the answer of the integral doesn't that mean that that series should converge to that answer?
     
  5. Mar 29, 2013 #4

    haruspex

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    Assuming no algebraic errors, yes. But it could be that the easiest way to sum the series is by reversing the steps to the integral and solving that.
     
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