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Integral of exp(-i*x^2)

  1. Mar 24, 2013 #1

    VVS

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    Hello!

    I am doing this purely out of curiousity.

    1. The problem statement, all variables and given/known data

    I am trying to integrate exp(-i*x^2) from -infinity to +infinity. Given that the integral from -infinity to infinity of exp(-x^2)=sqrt(pi).

    I typed it in Wolfram Alpha and I got (1/sqrt(2)-i*1/sqrt(2))*sqrt(pi).

    One can arrive at this solution by substituting the integral y=sqrt(i)x

    Then one gets (1/sqrt(2)-i/sqrt(2))*integral exp(-y^2) from -infinity to infinity

    BUT here is the catch. The limits have changed from -infinity to infinity to -infinity-i*infinity to +infinity+i*infinity.

    However if you just evaluate the integral from -infinity to +infinity you get the right answer.
    How can it be right? Isn't it mathematically inprecise? Or is there a mathematical theorem in complex analysis in which -infinity is the same the -i*infinity or something like that?

    I am really curious to know.

    Thank you
     
  2. jcsd
  3. Mar 24, 2013 #2

    D H

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    Staff Emeritus
    Science Advisor

    Expand [itex]\exp(-ix^2)[/itex] and you get [itex]\cos(x^2) - i\sin(x^2)[/itex]. Both [itex]\int_{-\infty}^{\infty} \cos(x^2)\,dx[/itex] and [itex]\int_{-\infty}^{\infty} \sin(x^2)\,dx[/itex] are well-defined. Google "Fresnel integral" for more info.
     
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