Integration of incomplete gamma function

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Hi,

I am interested in performing the following integration:

[tex]\int _{-\infty }^{\infty }\Gamma\left[k,\frac{x+u}{v}\right]e^{-\frac{(x-m)^2}{2\sigma ^2}}dx
[/tex].

I would appreciate anyone's help. I have been trying to do it in Mathematica but it runs out of time returning the same integral.

Thanks in advance.

Alex
 
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  • #2
jasonRF
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is [tex]k[/tex] and integer? If so, then the incomplete gamma function is a polynomial times an exponential:

http://en.wikipedia.org/wiki/Incomplete_gamma_function

so you just complete the square in the exponent and you have a finite sum of moments of a Gaussian, which are easy to look up

http://en.wikipedia.org/wiki/Normal_distribution

Of course this will be very messy to write out, and at the end you will have a finite sum of confluent hypergeometric functions (which may not be useful to you at all) but it is doable. Nicer forms may be possible, too.

For arbitrary [tex]k[/tex] this is really hard, I think.

good luck,

jason

jason
 
  • #3
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Thanks, your comment was helpful about the moment generating function of a Gaussian.

k is an integer in my case. I will give it a try.

Best Regards,

Alex
 
  • #4
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I tried to do what you suggested, but I am not sure how to proceed in completing the square. In case, [tex] k [/tex] is an integer we get the polynomial as you said times [tex] e^{-x+u/v} [/tex]. The other part of the integrand is a Gaussian exponent. Therefore, I do not see how to proceed especially since the moment of a Gaussian is obtained via the Fourier transform. The first exponent in my case is not complex.

Can I approximate the upper incomplete Gamma function [tex]\Gamma[k,\frac{x+u}{v}] [/tex] in some other way in order to then be able to perform the integration?

Thanks again.
 

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