# Improper definite integral $\int_0^\infty e^{-x^2}(x + k)^\alpha dx$

I am unsure how to calculate the following definite integral: $$\int_0^\infty e^{-x^2}(x + k)^{\alpha}dx,$$ where $k > 0$ and $\alpha$ is a real number. I tried integrating by parts and also a few things with the Gamma function, but I am not getting anywhere. Maybe some slick trick with contour integration? Any help will be highly appreciated, thanks!

• wh do you think it exists in closed form at all? – Alex Jan 13 '16 at 10:34
• You can express it with an infinite sum using binomial coefficient expansion (with non-natural $n$). – N74 Jan 13 '16 at 11:10
• @N74 Yes, you are right. But that might be very difficult to compute in closed form, I mean to sum up the series. I was hoping for something easier. – user304824 Jan 13 '16 at 12:11
• @Alex Correct, I don't know that for sure. That is part of my question. – user304824 Jan 13 '16 at 12:26
• For natural values of a, just expand the integrand using the binomial theorem, and switch the order of summation and integration. The result can be seen here. – Lucian Jan 13 '16 at 14:01