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What is $$\int\Gamma(x)dx$$ where $\Gamma(x)$ is the gamma function?

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    It is the antiderivative of the Gamma function, which doesn't go by any other name I am aware of (or wolfram alpha is aware of, more importantly https://www.wolframalpha.com/input/?i=integrate+Gamma(x)). – 2017-02-21
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    Computing the antiderivative of the Gamma function isn't of any importance whatsoever. – 2017-02-21
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    As spaceisdarkgreen stated, it doesn't really go by a special name. But you may be interested to know that $\int_{\gamma-i\infty}^{\gamma+i\infty}\Gamma(s)ds=\frac{2\pi i}{e},\;\;\gamma>0$. – 2017-02-21
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    $\log \Gamma(z)$ has a nice antiderivative $$\int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + \frac{z(1-z)}{2} + z \log \Gamma(z) - \log G(z+1)$$ http://math.stackexchange.com/questions/529205/evaluating-the-log-gamma-integral-int-0z-log-gamma-x-mathrm-dx-i – 2017-02-21
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    @ Staki42 Yes! I've certainly noticed myself how the primitive of the _gamma_ function _just seems not to crop-up __anywhere___! And yet, it's partly this very _inubiquity_ of it that makes it so fascinating! ¶ @ anyone else. I _think_ what the OP _probably_ meant - and it's _certainly_ what I mean in my recent post - it's not the integral of it in an _imaginary direction_; or some _convolution_ of it; of the logarithm or reciprocal of it (although the result in that connection is _fascinating_): it's the _plain straightforward real integral of the __function itself___ that is being queried! – 2018-12-02
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    @Zaid Alyafeai -- That primitive of the _logarithm_ of the gamma function is fascinating though - I neglected it the first time I looked at this post ... but now I see more clearly how lovely it is. I'm certainly going to keep a copy of that in my personal notes. I would be interested to know where you found it, or whence or how you got it. – 2018-12-03

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