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Integral involving sine and root

  1. Apr 25, 2009 #1
    1. The problem statement, all variables and given/known data

    \int_0^{\infty} sin(ax) / sqrt(x) dx

    2. Relevant equations



    3. The attempt at a solution
    I thought of using integration by parts, but that gets me nowhere. I'm not sure how to go about this problem.
     
  2. jcsd
  3. Apr 25, 2009 #2

    CompuChip

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    I have found a way of solving it but I don't know if you will be able to follow, because I don't know what integration techniques you have learned. It's perhaps a bit over-complicated, as well. But I did the integration in Mathematica, and what I got reminded me very much of a Gaussian integral.

    If you first do a substitution to u = x1/2 you get an integral over something like sin(a u2). By symmetry you can extend this integral to the entire real axis. Then note that what you want is precisely the imaginary part of ei a u2. Then you can use the result that the integral of exp(-a y2) over the real line is sqrt(pi / a) as long as Re(a) > 0 (technically, you will get Re(a) = 0 so you might need to take some limit, or perform a Wick rotation to make things rigorous) and you will get your answer.

    Anyone for an easier approach?
     
  4. Apr 26, 2009 #3
    Thanks. That's exactly what I needed.
     
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