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Complex Integral Problem

  1. Feb 8, 2010 #1
    1. The problem statement, all variables and given/known data

    By applying Cauchy's theorem to a suitable contour, prove that the integral of cos(ax2) = (pi/8a)1/2


    2. Relevant equations

    Cauchy's integral formula:

    http://en.wikipedia.org/wiki/Cauchy's_integral_formula

    3. The attempt at a solution

    I'm not sure where to go after what I've got scanned. I've tried a few other things, but I can't seem to get anywhere that looks like something I could solve.
     

    Attached Files:

  2. jcsd
  3. Feb 8, 2010 #2
    Hint: Consider the function exp(-a z^2).
     
  4. Feb 9, 2010 #3
    What I do know about eiaz^2 is that within 0 < argz < pi/4 it is analytic and
    eiaz^2-->0 as |z|-->infinity

    Could I expand that in terms of a Taylor series and use the method of residues to solve the integral? Would this be the correct expansion for cos(ax2):

    ex = 1 + x + x2/2! + x3/3!

    Taylor series for x = iaz2 added to the taylor series for x = -iaz2
    = 1/2(eiaz^2 + e-iaz^2) = 1/2[(1 + iaz2 - a2z4/2 - a3z6/6 +...) + (1 - iaz + a2z4/2 + a3z6/6 +...) = 1/2(2) = 1?

    and then would I express 1 as a complex exponential?

    Is that right?
     
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