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Consider the following integral:

$$\int\exp\left(ax+bx^2\right)x^{\eta}\mathrm{d}x$$

where $\eta\ge0$ is a real number, and $a$ and $b$ are also real numbers.

Can I express this integral in terms of some special functions, such as the incomplete Gamma function ($\Gamma(x;z_1,z_2) = \int_{z_1}^{z_2} t^{x-1}e^{-t}\mathrm{d}t$), or some other special function that is computable using standard numerical libraries?

a06e
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1 Answers1

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If you complete the square using some constant value you multiply & divide from outside the integral, you could write it as the $\eta$ moment of the normal distribution : $$\int x^\eta e^{(x-\mu)^2/2\sigma^2} dx$$

However to evaluate this you will go through the Gamma function which seems unavoidable.

This is also called the raw moment.

Note : This only works if $\eta\in \mathbb N$.

user88595
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  • What if $\eta$ is a real number? ... – a06e Jan 20 '14 at 19:24
  • I missed that out sorry. Incomplete Gamma function is the only way I see. – user88595 Jan 20 '14 at 19:43
  • Incomplete Gamma function is excellent... but do you have an expression in terms of the incomplete Gamma function? Because I don't see it. – a06e Jan 20 '14 at 19:59
  • No nice closed form. I only know hypergeometric series to evaluate it. Follow http://mathworld.wolfram.com/IncompleteGammaFunction.html – user88595 Jan 20 '14 at 20:10