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I like to know is there closed form of below integral

$$\Gamma[20,a]=\int^{\infty}_{a}t^{19}e^{-t}dt $$

I can find closed form when integral range is 0 from inf.

But it is not easy to search what i like to know.

Thank you!

Michael Burr
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Kim
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  • In general, no; this is the (upper) incomplete Gamma function: https://en.wikipedia.org/wiki/Incomplete_gamma_function . – Travis Willse Apr 28 '16 at 10:45
  • What goes wrong with the iterated integration by parts here (since the power on $t$ is an integer, one should eventually reach $\Gamma[0,a]$, which has a closed form). – Michael Burr Apr 28 '16 at 11:08

1 Answers1

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It is enough to use integration by parts or the following fact: if $p(x)$ is a polynomial, $$ \int p(x)e^{-x}\,dx = C-\left(p(x)+p'(x)+p''(x)+\ldots\right)e^{-x}. $$ So we have:

$$ \int_{a}^{+\infty}t^{19}e^{-t}\,dt = e^{-a}\sum_{k=0}^{19}\binom{19}{k}k! \,a^{19-k}.$$

Mark Viola
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Jack D'Aurizio
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