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Rieman Zeta

  1. Dec 11, 2008 #1
    1. The problem statement, all variables and given/known data
    Evaluate $P\int_{-\infty}^{\infty}\exp(imx^2}dx$, $m>0$.

    2. Relevant equations
    None


    3. The attempt at a solution
    Consider the following contour $C$ consisting of
    $C_1:=\{z=x+iy:y=0, x:-s \to s\}$
    $C_2:=\{z=x+iy:x=0, y:0 \to is\}$
    $C_3:=\{z=x+iy:x: s \to -s, y: is \to -is \}$
    $C_4:=\{z=x+iy:x=0: y:-is \to 0\}$

    This is where i am stuck
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 11, 2008 #2
    Sorry, i gave the wrong title
     
  4. Dec 11, 2008 #3
    Well, why was that contour chosen? Does the integral go to 0 on some segments?
     
  5. Dec 11, 2008 #4
    actually in C_2, x=s and in C_4, x=-s
     
  6. Dec 11, 2008 #5
    Ok, I worked so far as that it only remains to show that the integrals on C_2 and C_4 are zero. I suspect having to use an ML-bound but don't really know how to do this.
     
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