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Math Help - integration of multiplication of Gaussian and Lorentzian

  1. #1
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    Angry integration of multiplication of Gaussian and Lorentzian

    I need to evaluate very generally the following integration,

    \int_{-\infty}^{\infty}\frac{1}{x^{2}+a^{2}}e^{-{\frac{(x-x_{0})^{2}}{2\sigma^{2}}}}\mathrm{d}x

    Is there analytic expression for that? Thanks,
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  2. #2
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    Quote Originally Posted by bsmile View Post
    I need to evaluate very generally the following integration,

    \int_{-\infty}^{\infty}\frac{1}{x^{2}+a^{2}}e^{-{\frac{(x-x_{0})^{2}}{2\sigma^{2}}}}\mathrm{d}x

    Is there analytic expression for that? Thanks,
    Well WolframAlpha does not know what it is.

    CB
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    Mathematica does not give a solution, which well explains that a regular expression does not exist so far, but we are doing research, right? Calling for mathematicians to share their bright minds!
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  4. #4
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    Quote Originally Posted by bsmile View Post
    Mathematica does not give a solution, which well explains that a regular expression does not exist so far, but we are doing research, right? Calling for mathematicians to share their bright minds!
    Since there is an algorithm for doing integrals (Risch algorithm) that modern software does not return a closed for for an integral generally means that a closed for does not exist in terms of elementary functions.

    CB
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  5. #5
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    Quote Originally Posted by bsmile View Post
    Mathematica does not give a solution, which well explains that a regular expression does not exist so far, but we are doing research, right? Calling for mathematicians to share their bright minds!
    Quite right. What was the Captain thinking .....?

    In fact a solution does exist: There is a special function called the Fantastic F-function of the Second Kind that's exactly the answer to your problem. The properties of this function are left for you to explore as I don't think much literature on it exists at the moment.


    (The Fantastic F-function of the First Kind was suggested to someone quite a few years ago now as the answer to an unrelated problem. No doubt a function of the Third Kind will be suggested in the future to someone else).
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    Quote Originally Posted by mr fantastic View Post
    Quite right. What was the Captain thinking .....?

    In fact a solution does exist: There is a special function called the Fantastic F-function of the Second Kind that's exactly the answer to your problem. The properties of this function are left for you to explore as I don't think much literature on it exists at the moment.


    (The Fantastic F-function of the First Kind was suggested to someone quite a few years ago now as the answer to an unrelated problem. No doubt a function of the Third Kind will be suggested in the future to someone else).
    You are joking, right?! I am desperately looking for a closed form solution for that integral to save cpu time, and was glad to see the strange F-function, but a second thought realizes that it does not exist at the time being ....

    Quote Originally Posted by CaptainBlack View Post
    Since there is an algorithm for doing integrals (Risch algorithm) that modern software does not return a closed for for an integral generally means that a closed for does not exist in terms of elementary functions.

    CB

    Seems there are really very good mathematicians on this forum, , thanks for letting me know the Risch algorithm ...
    Last edited by mr fantastic; September 25th 2009 at 01:05 AM. Reason: Merged posts
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  7. #7
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    Quote Originally Posted by bsmile View Post
    You are joking, right?! I am desperately looking for a closed form solution for that integral to save cpu time, and was glad to see the strange F-function, but a second thought realizes that it does not exist at the time being ....
    Necessity is the mother of invention (of non-elementary functions).
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