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Math Help - Integral involving expoential - does it have a name?

  1. #1
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    Integral involving expoential - does it have a name?

    I'm working with this simple little integral,

    I_{l}=\int_{x_{i}}^{x_{f}}x^{l}\exp[-x^{2}]dx,

    and was wondering if it had a name. It seems so basic that I imagine there's a lot of literature written about it - I would like to know more about it.

    An observation: If the limits were 0 and infinity the solution would be related to gamma functions.


    I have found solutions for integer l (integrate by parts once, then note a recursion relation). Then general solutions can be written down; one for even l and one for odd l.

    e.g. for odd l

    I_{l}=\left[-\frac{1}{2}\exp\left(-x^{2}\right)\left\{ x^{l-1}+\sum_{n=1}^{N}\left(x^{(l-1)-2n}\prod_{m=1}^{n}\frac{(l-1)-2(m-1)}{2}\right)\right\} \right]_{x_{i}}^{x_{f}}

    I think that's correct - I'm just getting the feeling that it must have been solved a million times before...

    Any thoughts?

    Thanks in advance,
    Tom.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by astronomerroyal View Post
    I have found solutions for integer l
    What if  l=0 ?
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  3. #3
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    Quote Originally Posted by chiph588@ View Post
    What if  l=0 ?

    l=0? Error functions.

    For non-zero, positive integer l the integral reminds one of the moment equation for the gaussian. This is just one of the reasons I believe that huge tomes have already been written on this particular integral. However, I'm not finding the tomes (via google, wikipedia, colleagues etc.).
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