1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Math. physics

  1. Dec 5, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove

    [tex]\sqrt{\frac{2}{\pi}}\int^{\infty}_0x^{-\frac{1}{2}}\cos (xt)dx=t^{-\frac{1}{2}}[/tex]

    and use that to solve

    [tex]\int^{\infty}_0\cos y^2dy[/tex]

    Is this good way to try to prove?



    2. Relevant equations





    3. The attempt at a solution
    1. The problem statement, all variables and given/known data
    Multiplicate both sides with [tex]\cos x'tdt[/tex] and integrate from zero to [tex]\infty[/tex]

    [tex]\sqrt{\frac{2}{\pi}}\int^{\infty}_0dt\cos (x't)\int^{\infty}_0x^{-\frac{1}{2}}\cos (xt)dx=\int^{\infty}_0dt\cos (x't)t^{-\frac{1}{2}}=\sqrt{\frac{2}{\pi}}\int^{\infty}_0dxx^{-\frac{1}{2}}\int^{\infty}_0dt\cos (x't)\cos (xt)dx[/tex]
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 5, 2011 #2
    Suppose that we know

    [tex]\sqrt{\frac{2}{\pi}}\int^{\infty}_0\cos(xt)x^{-\frac{1}{2}}dx=t^{-\frac{1}{2}}[/tex]

    without proving. How to calculate then

    [tex]\int^{\infty}_0\cos x^2dx[/tex]
     
    Last edited: Dec 5, 2011
  4. Dec 5, 2011 #3
    you need to use residue calculus.
    if you can go to library look at hildebrand advanced calculus for applications, under the intended contours you will see how to use cauchy's principle and then you'll get gamma functions.
     
  5. Dec 5, 2011 #4
    For which part of problem. This is problem from Arfken, Weber.
     
  6. Dec 5, 2011 #5
    you can do the proof and also find part b when you understand the first part I assume.look hildebrand page 561 to be exact
     
  7. Dec 5, 2011 #6
    [tex]\int^{\infty}_0\frac{\cos x}{x^{1-m}}dx=\Gamma(m)\cos (\frac{m\pi}{2})[/tex]

    [tex]\int^{\infty}_0\frac{\cos x}{x^{1-\frac{1}{2}}}dx=\Gamma(\frac{1}{2})\cos (\frac{\frac{1}{2}\pi}{2})=\sqrt{\pi}\frac{\sqrt{2}}{2}=\sqrt{\frac{\pi}{2}}[/tex]

    I don't see solution :(
     
  8. Dec 5, 2011 #7
    you need to look at contour integration and use xt instead of x there. By using residue and appropriate contour you'll be able to find t^-1/2
     
  9. Dec 5, 2011 #8
    Ok. Thanks. And what then. When I prove first part, how can I calculate integral [tex]\int^{\infty}_0\cos x^2dx[/tex]?
     
  10. Dec 5, 2011 #9
  11. Dec 5, 2011 #10

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Take t=1 in what you have proven already then do a simple substitution to get rid of the sqrt.
     
  12. Dec 6, 2011 #11
    Thanks a lot! :)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Math. physics
  1. Tensor math (Replies: 1)

Loading...