Editing Incomplete gamma function (section)

==Evaluation formulae==

The lower gamma function can be evaluated using the power series expansion:<ref>{{Cite web|url=https://dlmf.nist.gov/8.11#ii|title=DLMF: §8.11 Asymptotic Approximations and Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions|website=dlmf.nist.gov}}</ref>
<math display="block">\gamma(s, z) = \sum_{k=0}^\infty \frac{z^s e^{-z} z^k}{s (s+1) \dots (s+k)}=z^s e^{-z}\sum_{k=0}^\infty\dfrac{z^k}{s^{\overline{k+1}}}</math>
where <math>s^{\overline{k+1}}</math> is the [[Falling and rising factorials|Pochhammer symbol]].

An alternative expansion is
<math display="block">\gamma(s,z)= \sum_{k=0}^\infty \frac{(-1)^k}{k!} \frac{z^{s+k}}{s+k}= \frac{z^s}{s} M(s, s+1,-z),</math>
where {{math|''M''}} is Kummer's [[confluent hypergeometric function]].

===Connection with Kummer's confluent hypergeometric function===

When the real part of {{mvar|z}} is positive,
<math display="block">\gamma(s,z) = s^{-1} z^s e^{-z} M(1,s+1,z)</math>
where <math display="block"> M(1, s+1, z) = 1 + \frac{z}{(s+1)} + \frac{z^2}{(s+1)(s+2)} + \frac{z^3}{(s+1)(s+2)(s+3)} + \cdots</math> has an infinite radius of convergence.

Again with [[confluent hypergeometric functions]] and employing Kummer's identity,
<math display="block">\begin{align}
\Gamma(s,z) &= e^{-z} U(1-s,1-s,z) = \frac{z^s e^{-z}}{\Gamma(1-s)} \int_0^\infty  \frac{e^{-u}}{u^s (z+u)} du \\
&= e^{-z} z^s U(1,1+s,z) = e^{-z} \int_0^\infty e^{-u} (z+u)^{s-1} du = e^{-z} z^s \int_0^\infty e^{-z u} (1+u)^{s-1} du.
\end{align}</math>

For the actual computation of numerical values, [[Gauss's continued fraction]] provides a useful expansion:
<math display="block">
\gamma(s, z) = \cfrac{z^s e^{-z}}{s - \cfrac{s z}{s+1 + \cfrac{z}{s+2 - \cfrac{(s+1)z}
{s+3 + \cfrac{2z}{s+4 - \cfrac{(s+2)z}{s+5 + \cfrac{3z}{s+6 - \ddots}}}}}}}.
</math>

This continued fraction converges for all complex {{mvar|z}}, provided only that {{mvar|s}} is not a negative integer.

The upper gamma function has the continued fraction<ref>Abramowitz and Stegun [http://www.math.sfu.ca/~cbm/aands/page_263.htm p.&nbsp;263, 6.5.31]</ref>
<math display="block">
\Gamma(s, z) = \cfrac{z^s e^{-z}}{z+\cfrac{1-s}{1 + \cfrac{1}{z + \cfrac{2-s}
{1 + \cfrac{2}{z+ \cfrac{3-s}{1+ \ddots}}}}}}
</math>
and{{Citation needed|date=February 2013}}
<math display="block">
\Gamma(s, z)= \cfrac{z^s e^{-z}}{1+z-s+ \cfrac{s-1}{3+z-s+ \cfrac{2(s-2)}{5+z-s+ \cfrac{3(s-3)} {7+z-s+ \cfrac{4(s-4)}{9+z-s+ \ddots}}}}}
</math>

===Multiplication theorem===
The following [[multiplication theorem]] holds true{{Citation needed|reason=Origin of statement unclear; also clarification of L-function would be helpful, possibly https://en.wikipedia.org/wiki/Laguerre_polynomials#Generalized_Laguerre_polynomials?|date=November 2024}}:
<math display="block">\Gamma(s,z) = \frac 1 {t^s} \sum_{i=0}^{\infty} \frac{\left(1-\frac 1 t \right)^i}{i!} \Gamma(s+i,t z)
= \Gamma(s,t z) -(t z)^s e^{-t z} \sum_{i=1}^{\infty} \frac{\left(\frac 1 t-1 \right)^i}{i} L_{i-1}^{(s-i)}(t z).</math>

===Software implementation===
The incomplete gamma functions are available in various of the [[computer algebra system]]s.

Even if unavailable directly, however, incomplete function values can be calculated using functions commonly included in [[spreadsheet]]s (and computer algebra packages).  In [[Microsoft Excel|Excel]], for example, these can be calculated using the [[gamma function]] combined with the [[gamma distribution]] function.
*The lower incomplete function: <math>  \gamma(s, x) </math> <code> = EXP(GAMMALN(s))*GAMMA.DIST(x,s,1,TRUE)</code>.
*The upper incomplete function: <math> \Gamma(s, x) </math> <code> = EXP(GAMMALN(s))*(1-GAMMA.DIST(x,s,1,TRUE))</code>.
These follow from the definition of the [[Gamma distribution#Cumulative distribution function|gamma distribution's cumulative distribution function]].

In [[Python (programming language)|Python]], the Scipy library provides implementations of incomplete gamma functions under {{code|scipy.special}}, however, it does not support negative values for the first argument. The function {{Code|gammainc}} from the mpmath library supports all complex arguments.