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Evaluate the integral.

  1. Sep 14, 2009 #1
    1. The problem statement, all variables and given/known data

    This is part of a much larger problem, however currently I am stuck at the following integral:

    [tex]

    \int_{-\infty}^{\infty}exp\left(k^2\left((\Delta x)^2-\frac{i \hbar t}{2m}\right ) + k(ix-2(\Delta x)^2 \bar{k}_x)\right)dk[/tex]

    Where obviously everything should be taken as a constant except the plane old k's.

    2. Relevant equations

    see (1) and (3)


    3. The attempt at a solution

    i tried to complete the square, followed by u/du substitution which yielded one of the messiest equations Ive ever seen. There is an integral I found in a table that looks promising:

    [tex]\int exp(-ax^2+bx+c)dx=\sqrt{\frac{\pi}{a}} exp\left(\frac{b^2}{4a}+c\right)[/tex]

    However this does not include the imaginary unit therefore I do not believe it is much use to me.

    Cookies for anyone who can get me started in the right direction.

    Thanks yall!

    IHateMayonnaise
     
    Last edited: Sep 15, 2009
  2. jcsd
  3. Sep 15, 2009 #2

    Avodyne

    User Avatar
    Science Advisor

    Your formula with a, b, & c is valid for complex a, b, & c as long as the real part of a is positive.

    I think the [itex]k^2(\Delta x)^2[/itex] term in the exponent has the wrong sign.
     
  4. Sep 15, 2009 #3
    Why is the formula valid only if the real part of a is positive? (Also, this is assuming that it is a definite integral from [itex]-\infty[/itex] to [itex]\infty[/itex]?)

    Good call on the [itex]k^2(\Delta x)^2[/itex] having the wrong sign! Thanks so much :)

    Edit: Also I made a mistake in that integral from the table; I originally put:

    [tex]\int exp(-ax^2+bx+c)dx=\sqrt{\frac{\pi}{a}} exp\left(\frac{b^2}{4ac}+c\right)[/tex]

    but I meant:

    [tex]\int exp(-ax^2+bx+c)dx=\sqrt{\frac{\pi}{a}} exp\left(\frac{b^2}{4a}+c\right)[/tex]
     
    Last edited: Sep 15, 2009
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