Editing Incomplete gamma function (section)

====Upper incomplete gamma function====

As for the '''upper incomplete gamma function''', a [[Holomorphic function|holomorphic]] extension, with respect to {{mvar|z}} or {{mvar|s}}, is given by<ref name="auto3"/>
<math display="block">\Gamma(s,z) = \Gamma(s) - \gamma(s, z)</math>
at points {{math|(''s'', ''z'')}}, where the right hand side exists. Since <math>\gamma</math> is multi-valued, the same holds for <math>\Gamma</math>, but a restriction to principal values only yields the single-valued principal branch of <math>\Gamma</math>.

When {{mvar|s}} is a non-positive integer in the above equation, neither part of the difference is defined, and a [[Limit of a function|limiting process]], here developed for {{math|''s'' → 0}}, fills in the missing values. [[Complex analysis]] guarantees [[holomorphic function|holomorphicity]], because <math>\Gamma(s,z)</math> proves to be [[Bounded function|bounded]] in a [[Neighbourhood (mathematics)|neighbourhood]] of that limit for a fixed {{mvar|z}}.

To determine the limit, the power series of <math>\gamma^*</math> at {{math|1=''z'' = 0}} is useful. When replacing <math>e^{-x}</math> by its power series in the integral definition of <math>\gamma</math>, one obtains (assume {{mvar|x}},{{mvar|s}} positive reals for now):
<math display="block">\gamma(s, x) = \int_0^x t^{s-1} e^{-t} \, dt = \int_0^x \sum_{k=0}^\infty \left(-1\right)^k \, \frac{t^{s+k-1}}{k!} \, dt = \sum_{k=0}^\infty \left(-1\right)^k \, \frac{x^{s+k}}{k!(s+k)} = x^s\,\sum_{k=0}^\infty \frac{(-x)^k}{k!(s+k)}</math>
or<ref name="auto1"/>
<math display="block">\gamma^*(s,x) = \sum_{k=0}^\infty \frac{(-x)^k}{k!\,\Gamma(s)(s+k)},</math>
which, as a series representation of the entire <math>\gamma^*</math> function, converges for all complex {{mvar|x}} (and all complex {{mvar|s}} not a non-positive integer).

With its restriction to real values lifted, the series allows the expansion:
<math display="block">\gamma(s, z) - \frac{1}{s} = - \frac{1}{s} + z^s\,\sum_{k=0}^\infty \frac{(-z)^k}{k!(s+k)} = \frac{z^s-1}{s} + z^s\, \sum_{k=1}^\infty \frac{\left(-z\right)^k}{k!(s+k)},\quad \Re(s) > -1, \,s \ne 0.</math>

When {{math|''s'' → 0}}:<ref>[[Gamma function#General|see last eq.]]</ref>
<math display="block">\frac{z^s-1}{s} \to \ln(z),\quad \Gamma(s) - \frac{1}{s} = \frac{1}{s} - \gamma + O(s) - \frac{1}{s} \to -\gamma,</math>
(<math>\gamma</math> is the [[Euler–Mascheroni constant]] here), hence,
<math display="block">\Gamma(0,z) = \lim_{s \to 0}\left(\Gamma(s) - \tfrac{1}{s} - (\gamma(s, z) - \tfrac{1}{s})\right) = -\gamma - \ln(z) - \sum_{k=1}^\infty \frac{(-z)^k}{k\,(k!)}</math>
is the limiting function to the upper incomplete gamma function as {{math|''s'' → 0}}, also known as the [[exponential integral]] {{nowrap|<math>E_1(z)</math>.}}<ref>{{Cite web|url=https://dlmf.nist.gov/8.4|title=DLMF: §8.4 Special Values ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions|website=dlmf.nist.gov}}</ref>

By way of the recurrence relation, values of <math>\Gamma(-n, z)</math> for positive integers {{mvar|n}} can be derived from this result,<ref>{{Cite web|url=http://dlmf.nist.gov/8.4.E15|title = DLMF: 8.4 Special Values}}</ref>
<math display="block">\Gamma(-n, z) = \frac{1}{n!} \left(\frac{e^{-z}}{z^n} \sum_{k = 0}^{n - 1} (-1)^k (n - k - 1)! \, z^k + \left(-1\right)^n \Gamma(0, z)\right)</math>
so the upper incomplete gamma function proves to exist and be holomorphic, with respect both to {{mvar|z}} and {{mvar|s}}, for all {{mvar|s}} and {{math|''z'' ≠ 0}}.

<math>\Gamma(s, z)</math> is:
* [[Entire function|entire]] in {{mvar|z}} for fixed, positive integral {{mvar|s}};
* multi-valued [[Holomorphic function|holomorphic]] in {{mvar|z}} for fixed {{mvar|s}} non zero and not a positive integer, with a [[branch point]] at {{math|1=''z'' = 0}};
* equal to <math>\Gamma(s)</math> for {{mvar|s}} with positive real part and {{math|1=''z'' = 0}} (the limit when <math>(s_i,z_i) \to (s, 0)</math>), but this is a continuous extension, not an [[analytic continuation|analytic one]] ('''does not''' hold for real {{math|''s'' < 0}}!);
* on each branch [[Entire function|entire]] in {{mvar|s}} for fixed {{math|''z'' ≠ 0}}.