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Math Help - Integral(1/t * e^(t- 1/t))

  1. #1
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    Integral(1/t * e^(t- 1/t))

    Hello, I have problems to show that

    \int_0^{\infty} e^{-ct}\sqrt{\frac{1}{2\pi t}}e^{-(x-y)^2\frac{1}{2t}} dt=\frac{1}{\sqrt{2c}}e^{-\sqrt{2c}|x-y|}

    If one omits the constants then essentially on has to calculate

    \int_0^{\infty}\sqrt{\frac{1}{t}}e^{-t-\frac{1}{t}} dt.

    I can't see, if and how substitution or partial integration could help.

    Maybe the residual theorem could help? But I dont see directly how...

    Do you have an idea how to calculate this integral? I would be glad, if somebody has.

    best regards,
    slabic
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  2. #2
    Moo
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    Re: Integral(1/t * e^(t- 1/t))

    Hello,

    It looks like something from probability... Is it the original problem ?
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  3. #3
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    Re: Integral(1/t * e^(t- 1/t))

    yes, the background is a prove of a specific semigroup-representation. This topic can be connected to markov-transition kernels.
    To prove this representation, the only step left, is the problem above.


    But I think I'm on a good way with completing the exponent to a square and after substitution one gets an expression which is almost the gammafunction.
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