# Integration of incomplete gamma function

## Main Question or Discussion Point

Hi,

I am interested in performing the following integration:

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

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.

Alex

Last edited:

jasonRF
Gold Member
is $$k$$ 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 $$k$$ this is really hard, I think.

good luck,

jason

jason

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

Best Regards,

Alex

I tried to do what you suggested, but I am not sure how to proceed in completing the square. In case, $$k$$ is an integer we get the polynomial as you said times $$e^{-x+u/v}$$. 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 $$\Gamma[k,\frac{x+u}{v}]$$ in some other way in order to then be able to perform the integration?

Thanks again.