Editing Incomplete gamma function (section)

===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>