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Math Help - Integral Evaluation

  1. #1
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    Integral Evaluation

    I need to evaluate the following integral
    \int{\frac{e^{(-at-bt^2)}}{t}}dt
    It seems there is no closed from expression for this integral. But if we do the power series expansion for either e^{-at} or e^{-bt^2}, then the closed form is possible but with an infinite power series. Kindly suggest if any alternative is available?
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  2. #2
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    Quote Originally Posted by rpmatlab View Post
    But if we do the power series expansion for either e^{-at} or e^{-bt^2}, then the closed form is possible but with an infinite power series.
    Infinite power series does not constitute as a closed form though, right?
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by rpmatlab View Post
    I need to evaluate the following integral \int{\frac{e^{(-at-bt^2)}}{t}}dt

    What is the context of that integral?. Perhaps you only need to find the integral from 0 to +\infty .
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  4. #4
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    Hi Resilient, thank you for pointing out that infinite series will not be considered as Closed form. May be after truncating the series appropriately, we will get approximate closed form expression.
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Resilient View Post
    Infinite power series does not constitute as a closed form though, right?
    But it does constitute a symbolic representation that can be further manipulated, which is what I beleive is required in this case.

    CB
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  6. #6
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    The limits of the integral is from \sqrt{c} to \inf, where c is a non negative constant. After solving the integral through series expansion, I got a series of the form
    \sum_{k=0}^{\inf}(\frac{-b}{a^2})^k \frac{1}{k!} \Gamma(2k,b\sqrt{c})

    I want to truncate the series. But the range of \frac{b}{a^2} is 10^{-5} to 10^{5}. Also \Gamma(2k,b\sqrt{c}) is increasing with k. The series seems to be divergent for \frac{b}{a^2}\geq 1. How to truncate this type of series?

    Is this correct way to solve this integral or any other alternative is possible?


    If there is no other alternative to solve this kind of integral, can you please suggest a method to truncate the divergent series.
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