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Evaluating Fresnel Integrals

  1. Feb 1, 2016 #1
    1. The problem statement, all variables and given/known data
    Evaluate the following integrals C = 0inf∫cos(x2) dx and S = 0inf∫sin(x2) dx

    2. Relevant equations

    Hint: use Euler formula to write the integral for F = C + iS. Square the integral and evaluate it in polar coordinates. Temporary add a convergence factor.

    Answer: C = S = sqrt(pi/8)

    3. The attempt at a solution

    (Abbreviated form)

    F2 = 1/4 0inf∫ eix^2dx 0inf∫eiy^2dy

    F2 = pi/2 0inf∫ eir^2 r dr

    F2 = pi/4 0inf∫eiudu

    Now I think the convergence factor comes in here, but I am not entirely sure how that works.

    Thanks!
     
    Last edited: Feb 1, 2016
  2. jcsd
  3. Feb 1, 2016 #2

    Ray Vickson

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    You can look at
    [tex] J_r = \int_0^{\infty} e^{-ru} e^{iu} \, du,[/tex]
    where ##r > 0##. Then take the limit as ##r \to 0##.
     
  4. Feb 1, 2016 #3
    Great, so that just comes to i, correct?

    So,

    F^2=π/4 eiπ/2 --> F = ei/4sqrt(π/4)

    How do I get to S and C from here? (I know if I evaluate the product of sqrt(π/4) and sin(π/4)2 it spits out sqrt(π/8). Why does this work?)
     
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