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Math Help - \int_{k}^{\infty} e^{-a^{2}} dy (gauss int. w. finite lim.)

  1. #1
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    \int_{k}^{\infty} e^{-a^{2}} dy (gauss int. w. finite lim.)

    I am doing a calculation of overlap between wavefunctions in a physics thesis, however this integral puzzles me:

    \int_{-L/2+\kappa a^{2}}^{\infty} e^{-\frac{1}{2a^{2}}y^{2}} dy

    From a physic point of view it is very important that the integral is from -L/2+\kappa a^{2} and not from -\infty. I am awere that is is impossible to write down an antiderivative of the function being integrated however, there must be some way to deal with the integral.

    How do I calculate this integral (as a function of momentum \kappa)?
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  2. #2
    Super Member wingless's Avatar
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    D=-L/2+\kappa a^{2}

    \int_{D}^{\infty} e^{-\frac{y^2}{2a^{2}}} dy

    \int_{D}^{\infty} e^{-\left ( \frac{y}{a\sqrt{2}}\right )^2} dy

    Let x =\frac{y}{a\sqrt{2}}. Then dy = a\sqrt{2}~dx.

    The bounds are now from \frac{D}{a\sqrt{2}} to infinity.

    It becomes \int_{\frac{-L/2+\kappa a^{2}}{a\sqrt{2}}}^{\infty}e^{-x^2}~dx.

    \int_{\frac{-L/2+\kappa a^{2}}{a\sqrt{2}}}^{\infty}e^{-x^2}~dx = \int_{0}^{\infty}e^{-x^2}~dx - \int_{0}^{\frac{-L/2+\kappa a^{2}}{a\sqrt{2}}}e^{-x^2}~dx

    \int_{0}^{\infty}e^{-x^2}~dx is the Gaussian integral and \frac{\sqrt{\pi}}{2}

    \frac{\sqrt{\pi}}{2} - \int_{0}^{\frac{-L/2+\kappa a^{2}}{a\sqrt{2}}}e^{-x^2}~dx

    \frac{\sqrt{\pi}}{2} - \frac{\sqrt{\pi}}{2} \mbox{Erf}(\frac{-L/2+\kappa a^{2}}{a\sqrt{2}})

    You have to use an approximation of the error function here.
    An approximation of Erf is: |\mbox{Erf}(x)| = \sqrt{1-e^{-(2x/\sqrt{\pi})^2}}

    Another one (and a better one, I think) is here: Error function - Wikipedia, the free encyclopedia
    Last edited by wingless; May 30th 2008 at 05:48 AM.
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  3. #3
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    wingless, thanks a lot. I really appreciate it.

    I've never come across error functions in this context if ever at all. Seems like I have only done easy -inf to inf gaussians in electro class.
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  4. #4
    Super Member wingless's Avatar
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    Quote Originally Posted by AA1983 View Post
    wingless, thanks a lot. I really appreciate it.

    I've never come across error functions in this context if ever at all. Seems like I have only done easy -inf to inf gaussians in electro class.
    Not at all.. If you are looking for more about the error function, you can see the wikipedia thread
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