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

**slabic** · May 13th 2012, 04:22 AM

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

**Moo** · May 13th 2012, 11:39 PM

Hello,

It looks like something from probability... Is it the original problem ?

**slabic** · May 14th 2012, 10:08 AM

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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