Editing Incomplete gamma function (section)

=====Holomorphic extension=====
Repeated application of the recurrence relation for the '''lower incomplete gamma''' function leads to the [[power series]] expansion: <ref name="auto2">{{Cite web|url=https://dlmf.nist.gov/8.8|title=DLMF: §8.8 Recurrence Relations and Derivatives ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions|website=dlmf.nist.gov}}</ref>
<math display="block">\gamma(s, x) = \sum_{k=0}^\infty \frac{x^s e^{-x} x^k}{s(s+1)\cdots(s+k)} = x^s \, \Gamma(s) \, e^{-x} \sum_{k=0}^\infty\frac{x^k}{\Gamma(s+k+1)}.</math>
Given the rapid growth in [[Absolute value#Complex numbers|absolute value]] of {{math|Γ(''z'' + ''k'')}} when {{math|''k'' → ∞}}, and the fact that the [[Reciprocal Gamma function|reciprocal of {{math|Γ(''z'')}}]] is an [[entire function]], the coefficients in the rightmost sum are well-defined, and locally the sum [[Uniform convergence|converges uniformly]] for all complex {{mvar|s}} and {{mvar|x}}. By a theorem of [[Weierstrass]],<ref name="class notes">{{cite web |url=http://www.math.washington.edu/~marshall/math_534/Notes.pdf |title=Complex Analysis | work=Math 534 |date= Autumn 2009 | author = Donald E. Marshall |publisher=University of Washington |type=student handout |access-date=2011-04-23 |url-status=dead |archive-url=https://web.archive.org/web/20110516005152/http://www.math.washington.edu/~marshall/math_534/Notes.pdf |archive-date=2011-05-16 |at= Theorem 3.9 on p.56}}</ref> the limiting function, sometimes denoted as {{nowrap|<math>\gamma^*</math>,}}<ref name="auto1">{{Cite web|url=https://dlmf.nist.gov/8.7|title=DLMF: §8.7 Series Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions|website=dlmf.nist.gov}}</ref>
<math display="block">\gamma^*(s, z) := e^{-z}\sum_{k=0}^\infty\frac{z^k}{\Gamma(s+k+1)}</math>
is [[Entire function|entire]] with respect to both {{mvar|z}} (for fixed {{mvar|s}}) and {{mvar|s}} (for fixed {{mvar|z}}),<ref name="auto3"/> and, thus, holomorphic on {{math|'''C''' × '''C'''}} by [[Hartog's theorem]].<ref>{{cite web|author=Paul Garrett|url=https://www-users.cse.umn.edu/~garrett/m/complex/hartogs.pdf|title=Hartogs' Theorem: separate analyticity implies joint|website=cse.umn.edu|access-date=21 December 2023}}</ref> Hence, the following ''decomposition''<ref name="auto3"/>
<math display="block">\gamma(s,z) = z^s \, \Gamma(s) \, \gamma^*(s,z),</math>
extends the real lower incomplete gamma function as a [[holomorphic function]], both jointly and separately in {{mvar|z}} and {{mvar|s}}. It follows from the properties of <math>z^s</math> and the [[Gamma function|Γ-function]], that the first two factors capture the [[Mathematical singularity|singularities]] of <math>\gamma(s,z)</math> (at {{math|1=''z'' = 0}} or {{mvar|s}} a non-positive integer), whereas the last factor contributes to its zeros.