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Integrate exp(-(z-ia)^2) from z = - infinity to z = infinity

  1. Mar 17, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove that [itex]\int^{∞}_{-∞}[/itex] exp(-(z-ia)2)dz = √∏ for all real a.

    2. Relevant equations



    3. The attempt at a solution

    If I use the substitution x = z-ia then dz = dx and if I use the limits x = -∞ to x = ∞ I get the correct answer. However, I do not know how to justify leaving the limits the same or if it is even ok?
     
  2. jcsd
  3. Mar 17, 2012 #2
  4. Mar 17, 2012 #3
    No it's not ok but you can still solve it by making that substitution [itex]u=z-ia[/itex], just change the limits on the integral:

    [tex]\int_{-\infty}^{\infty}e^{-(z-ia)^2}dz=\int_{-\infty-ia}^{\infty+ia} e^{-u^2}du[/tex]

    Now, that integrand is analytic so independent of path so that I can go from [itex]-\infty-ia[/itex] up to the point [itex]-\infty[/itex], go down the real axis to [itex]\infty[/itex] then up to the point [itex]\infty+ia[/itex]. The two vertical legs are zero because of the negative exponent so that we're left with just the ordinary gaussian integral which is [itex]\sqrt{\pi}[/itex]
     
    Last edited: Mar 17, 2012
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