Talk:Incomplete gamma function

Typographical discrepancy
The Incomplete gamma function article uses \mathrm{e} throughout. That disagrees with the article on the mathematical constant e, the article on the gamma function, and with the mathematical literature as a whole. "\mathrm{e}" should be replaced with "e" throughout.Getthebasin (talk) 17:05, 8 September 2020 (UTC)

I have made the replacement.Getthebasin (talk) 17:19, 8 September 2020 (UTC)

Relation between types?
Question: What is the relation between two definitions of incomplete Gamma functions ? Are they the same ? It seems not. — Preceding unsigned comment added by 169.229.120.24 (talk) 20:59, 6 May 2006 (UTC)


 * They're not the same, but the relationship between them is


 * $$ \gamma(a,x) + \Gamma(a,x) = \Gamma(a)\, $$. Evil saltine 21:12, 6 May 2006 (UTC)


 * It would seem that the definition of the two gammas here is different from that on the Poisson distribution page. There the cumulative distribution is said to be related to capital Gamma; but by this definition, it would be lower case Gamma.  —Preceding unsigned comment added by YouRang? (talk • contribs) 16:17, 8 August 2008 (UTC)

lower incomplete gamma function for small x'es
The current article states

$$\gamma(a,x) = \int_0^x t^{a-1}\,e^{-t}\,dt .\,\!$$ and $$ \frac{\gamma(a,x)}{x^a} \rightarrow 1 \quad \mathrm{as\ } x \rightarrow 0, \,$$ shouldn't it maybe be something like, - say - $$ \frac{\gamma(a,x)}{x^a} \rightarrow a^{-1} \quad \mathrm{as\ } x \rightarrow 0, \,$$ I have just created my wikipedia-account today, so Im not really sure how this stuff works, but wouldn't you agree that an extra a in the above formula makes sense? (b.t.w. I didn't place that retangular dashed boxed in my post on purpose)

Robert (talk) 21:59, 3 December 2008 (UTC)

There is no $$a$$ in the formula. As mentioned in the section on derivatives, $$\partial_x \gamma(a,x) = x^{a-1}e^{-x}$$, and so as x goes to zero, $$ \frac{\gamma(a,x)}{x^a} \approx e^{-x} \rightarrow 1. $$

Petrelharp (talk) 20:33, 12 November 2009 (UTC)

derivatives
I have just added some material concerning derivatives of the (upper) incomplete gamma function with respect to its argument(s). I decided to be bold and also added some material indicating how and why this is useful. This material has been in the literature for more than 20 years. I thought it was high time it was added. —Preceding unsigned comment added by 171.71.55.135 (talk) 22:38, 12 January 2009 (UTC)

great! very useful. Petrelharp (talk) 20:33, 12 November 2009 (UTC)


 * Thank you! -- Tony 69.106.255.20 (talk) 06:33, 25 January 2010 (UTC)

asymptotic formula?
I wonder what the restriction |arg s| < 1.5*pi in the asymptotic formula of Gamma(s,z) really means and I suspect the constraints of the formula are not correctly cited here. If one takes arg running from -pi to pi (the principle value, as defined here http://en.wikipedia.org/wiki/Argument_%28complex_analysis%29), then the restriction always holds, and can simply be omitted. To my knowledge, formulas related to the incomplete gamma function often hold in a (complex) half plane only, so some kind of constraint might be necessary here, but it is certainly not the one mentioned here. I got some books (Temme, Cruyt, Abramowitz/Stegun and others) that cover the incomplete gamma function, and I'm going to look that up these days.

Wolf Lammen —Preceding unsigned comment added by 85.183.23.210 (talk) 09:23, 15 February 2011 (UTC)

http://functions.wolfram.com/GammaBetaErf/Gamma2/06/02/02/0003/

The link indicates the correct restrictions are (since the formula holds for complex values as well, choose z as variable name): $$\Gamma(s,z) \sim z^{s-1} e^{-z} \, \sum_{k=0} \frac {\Gamma(s)} {\Gamma(s-k)} z^{-k}$$ as an asymptotic series where $$|z| \to \infty$$.

Actually, replacing $$\frac {\Gamma(s)} {\Gamma(s-k)}$$ by the falling factorial $$(-1)^k(1-s)_k$$, as it is done in the mentioned link, is recommended, as it avoids the poles of the gamma function at negative integers. —Preceding unsigned comment added by 85.183.23.210 (talk) 10:21, 15 February 2011 (UTC)

Wolf Lammen —Preceding unsigned comment added by 85.183.23.210 (talk) 10:08, 15 February 2011 (UTC)

Many formulas related to the incomplete gamma functions are found here: http://dlmf.nist.gov/8 The asymptotic formula above is found (slightly rewritten) here: http://dlmf.nist.gov/8.11. And the constraint $$|\arg\, z| < 1.5 * \pi$$ turns out to describe the sector where the formula is valid. The sector covers more than a full circle, because the formula is even partly valid in the branches adjacent to the principal branch of $$\Gamma(s, z)$$.

Wolf Lammen --85.183.23.210 (talk) 10:41, 21 February 2011 (UTC)

Eliminate the animations at the beginning of the article
The figures at the beginning of the article are hard to interpret. Moreover, the fact that they are animated makes them very annoying, like a banner from the 90's — Preceding unsigned comment added by 129.132.224.85 (talk) 15:29, 11 March 2015 (UTC)


 * Agreed. What are those figures even supposed to be showing? Caption info is inadequate. I've gotten rid of them. Something informative would be helpful. Isambard Kingdom (talk) 16:27, 17 July 2015 (UTC)

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