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I Gaussian integration for complex phase

  1. Oct 17, 2016 #1
    I would like to prove that

    ##\displaystyle{\int dx'\ \frac{1}{\sqrt{AB}}\exp\bigg[i\frac{(x''-x')^{2}}{A}\bigg]\exp\bigg[i\frac{(x'-x)^{2}}{B}\bigg]=\frac{1}{\sqrt{A+B}}\exp\bigg[i\frac{(x''-x)^{2}}{A+B}\bigg]}##

    Is there an easy way to do this integration that does not involve squaring the brackets?
     
  2. jcsd
  3. Oct 18, 2016 #2

    stevendaryl

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    Hmm. Completing the square would seem to me by far to be the easiest way to do it. Why do you want a different way?

    A second comment: Are you sure that is correct? Gaussian integrals usually involve factors of [itex]\sqrt{2\pi}[/itex]
     
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