Editing Incomplete gamma function (section)

====Lower incomplete gamma function====

=====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.

=====Multi-valuedness=====
The [[complex logarithm]] {{math|1=log ''z'' = log {{abs|''z''}} + ''i'' arg ''z''}} is determined up to a multiple of {{math|2''πi''}} only, which renders it [[Multi-valued function|multi-valued]]. Functions involving the complex logarithm typically inherit this property. Among these are the [[Exponentiation#nth roots of a complex number|complex power]], and, since {{math|''z''<sup>''s''</sup>}} appears in its decomposition, the {{math|γ}}-function, too.

The indeterminacy of multi-valued functions introduces complications, since it must be stated how to select a value. Strategies to handle this are:
* (the most general way) replace the domain {{math|'''C'''}} of multi-valued functions by a suitable manifold in {{math|'''C''' × '''C'''}} called [[Riemann surface]]. While this removes multi-valuedness, one has to know the theory behind it;<ref>{{cite web|author=C. Teleman
|url=http://math.berkeley.edu/~teleman/math/Riemann.pdf|title=Riemann Surfaces|website=berkeley.edu|access-date=21 December 2023}}</ref>  
* restrict the domain such that a multi-valued function decomposes into separate single-valued [[Branch point|branches]], which can be handled individually.

The following set of rules can be used to interpret formulas in this section correctly. If not mentioned otherwise, the following is assumed:

======Sectors======
Sectors in {{math|'''C'''}} having their vertex at {{math|1=''z'' = 0}} often prove to be appropriate domains for complex expressions. A sector {{mvar|D}} consists of all complex {{mvar|z}} fulfilling {{math|''z'' ≠ 0}} and {{math|''α'' − ''δ'' < arg ''z'' < ''α'' + ''δ''}} with some {{mvar|α}} and {{math|0 < ''δ'' ≤ ''π''}}. Often, {{mvar|α}} can be arbitrarily chosen and is not specified then. If {{mvar|δ}} is not given, it is assumed to be {{pi}}, and the sector is in fact the whole plane {{math|'''C'''}}, with the exception of a half-line originating at {{math|1=''z'' = 0}} and pointing into the direction of {{math|−''α''}}, usually serving as a [[Branch cut#Branch cuts|branch cut]]. Note: In many applications and texts, {{mvar|α}} is silently taken to be 0, which centers the sector around the positive real axis.

======Branches======
In particular, a single-valued and holomorphic logarithm exists on any such sector D having its imaginary part bound to the range {{open-open|''α'' − ''δ'', ''α'' + ''δ''}}. Based on such a restricted logarithm, {{math|''z''<sup>''s''</sup>}} and the incomplete gamma functions in turn collapse to single-valued, holomorphic functions on {{mvar|D}} (or {{math|'''C'''×''D''}}), called branches of their multi-valued counterparts on D. Adding a multiple of {{math|2''π''}} to {{mvar|α}} yields a different set of correlated branches on the same set {{mvar|D}}. However, in any given context here, {{mvar|α}} is assumed fixed and all branches involved are associated to it. If {{math|{{abs|''α''}} < ''δ''}}, the branches are called [[principal branch|principal]], because they equal their real analogues on the positive real axis. Note: In many applications and texts, formulas hold only for principal branches.

======Relation between branches======
The values of different branches of both the complex power function and the lower incomplete gamma function can be derived from each other by multiplication of <math>e^{2\pi iks}</math>,<ref name="auto3"/> for {{mvar|k}} a suitable integer.

=====Behavior near branch point=====
The decomposition above further shows, that γ behaves near {{math|1=''z'' = 0}} [[asymptotic]]ally like:
<math display="block">\gamma(s, z) \asymp z^s \, \Gamma(s) \, \gamma^*(s, 0) = z^s \, \Gamma(s)/\Gamma(s+1) = z^s/s.</math>

For positive real {{mvar|x}}, {{mvar|y}} and {{mvar|s}}, {{math|''x''<sup>''y''</sup>/y → 0}}, when {{math|(''x'', ''y'') → (0, ''s'')}}. This seems to justify setting {{math|1=''γ''(''s'', 0) = 0}} for real {{math|''s'' > 0}}. However, matters are somewhat different in the complex realm. Only if (a) the real part of {{mvar|s}} is positive, and (b) values {{math|''u''<sup>''v''</sup>}} are taken from just a finite set of branches, they are guaranteed to converge to zero as {{math|(''u'', ''v'') → (0, ''s'')}}, and so does {{math|''γ''(''u'', ''v'')}}. On a single [[branch point|branch]] of {{math|''γ''(''b'')}} is naturally fulfilled, so '''there''' {{math|1=''γ''(''s'', 0) = 0}} for {{mvar|s}} with positive real part is a [[Continuous function|continuous limit]]. Also note that such a continuation is by no means an [[analytic continuation|analytic one]].

=====Algebraic relations=====
All algebraic relations and differential equations observed by the real {{math|''γ''(''s'', ''z'')}} hold for its holomorphic counterpart as well. This is a consequence of the identity theorem, stating that equations between holomorphic functions valid on a real interval, hold everywhere. In particular, the recurrence relation <ref name="auto2"/> and {{math|1=''∂γ''(''s'', ''z'')/''∂z'' = ''z''<sup>''s''−1</sup> ''e''<sup>−''z''</sup>}} <ref name="auto2"/> are preserved on corresponding branches.

=====Integral representation=====
The last relation tells us, that, for fixed {{mvar|s}}, {{mvar|γ}} is a [[Primitive function|primitive or antiderivative]] of the holomorphic function {{math|''z''<sup>''s''−1</sup> ''e''<sup>−''z''</sup>}}. Consequently, for any complex {{math|''u'', ''v'' ≠ 0}},
<math display="block">\int_u^v t^{s-1}\,e^{-t}\, dt = \gamma(s,v) - \gamma(s,u)</math>
holds, as long as the [[Line integral|path of integration]] is entirely contained in the domain of a branch of the integrand. If, additionally, the real part of {{mvar|s}} is positive, then the limit {{math|''γ''(''s'', ''u'') → 0}} for {{math|''u'' → 0}} applies, finally arriving at the complex integral definition of {{math|''γ''}}<ref name="auto3"/>
<math display="block">\gamma(s, z) = \int_0^z t^{s-1}\,e^{-t}\, dt, \, \Re(s) > 0. </math>

Any path of integration containing 0 only at its beginning, otherwise restricted to the domain of a branch of the integrand, is valid here, for example, the straight line connecting {{math|0}} and {{mvar|z}}.

=====Limit for {{math|''z'' → +∞}}=====

======Real values======
Given the integral representation of a principal branch of {{math|''γ''}}, the following equation holds for all positive real {{mvar|s}}, {{mvar|x}}:<ref>{{Cite web|url=https://dlmf.nist.gov/5.2|title=DLMF: §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function|website=dlmf.nist.gov}}</ref>
<math display="block">\Gamma(s) = \int_0^\infty t^{s-1}\,e^{-t}\, dt = \lim_{x \to \infty} \gamma(s, x)</math>

======''s'' complex======
This result extends to complex {{mvar|s}}. Assume first {{math|1 ≤ Re(''s'') ≤ 2}} and {{math|1 < ''a'' < ''b''}}. Then
<math display="block">\left|\gamma(s, b) - \gamma(s, a)\right| \le \int_a^b \left|t^{s-1}\right| e^{-t}\, dt = \int_a^b t^{\Re s-1} e^{-t}\, dt \le \int_a^b t e^{-t}\, dt</math>
where<ref>{{Cite web|url=https://dlmf.nist.gov/4.4|title=DLMF: §4.4 Special Values and Limits ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions|website=dlmf.nist.gov}}</ref>
<math display="block">\left|z^s\right| = \left|z\right|^{\Re s} \, e^{-\Im s\arg z}</math>
has been used in the middle. Since the final integral becomes arbitrarily small if only {{mvar|a}} is large enough, {{math|''γ''(''s'', ''x'')}} converges uniformly for {{math|''x'' → ∞}} on the strip {{math|1 ≤ Re(s) ≤ 2}} towards a holomorphic function,<ref name="class notes" /> which must be Γ(s) because of the identity theorem. Taking the limit in the recurrence relation {{math|1=''γ''(''s'', ''x'') = (''s'' − 1) ''γ''(''s'' − 1, ''x'') − ''x''<sup>''s'' − 1</sup> ''e''<sup>−''x''</sup>}} and noting, that lim {{math|1=''x''<sup>''n''</sup> ''e''<sup>−''x''</sup> = 0}} for {{math|''x'' → ∞}} and all {{mvar|n}}, shows, that {{math|''γ''(''s'', ''x'')}} converges outside the strip, too, towards a function obeying the recurrence relation of the Γ-function. It follows
<math display="block">\Gamma(s) = \lim_{x \to \infty} \gamma(s, x)</math>
for all complex {{mvar|s}} not a non-positive integer, {{mvar|x}} real and {{math|''γ''}} principal.

======Sectorwise convergence======
Now let {{mvar|u}} be from the sector {{math|{{abs|arg ''z''}} < ''δ'' < ''π''/2}} with some fixed {{mvar|δ}} ({{math|1=''α'' = 0}}), {{math|''γ''}} be the principal branch on this sector, and look at
<math display="block">\Gamma(s) - \gamma(s, u) = \Gamma(s) - \gamma(s, |u|) + \gamma(s, |u|) - \gamma(s, u).</math>

As shown above, the first difference can be made arbitrarily small, if {{math|{{abs|''u''}}}} is sufficiently large. The second difference allows for following estimation:
<math display="block">\left|\gamma(s, |u|) - \gamma(s, u)\right| \le \int_u^{|u|} \left|z^{s-1} e^{-z}\right| dz = \int_u^{|u|} \left|z\right|^{\Re s - 1} \, e^{-\Im s\,\arg z} \, e^{-\Re z} \, dz,</math>
where we made use of the integral representation of {{math|''γ''}} and the formula about {{math|{{abs|''z''<sup>''s''</sup>}}}} above. If we integrate along the arc with radius {{math|1=''R'' = {{abs|''u''}}}} around 0 connecting {{mvar|u}} and {{math|{{abs|''u''}}}}, then the last integral is
<math display="block">\le R \left|\arg u\right| R^{\Re s - 1}\, e^{\Im s\,|\arg u|}\,e^{-R\cos\arg u} \le \delta\,R^{\Re s}\,e^{\Im s\,\delta}\,e^{-R\cos\delta} = M\,(R\,\cos\delta)^{\Re s}\,e^{-R\cos\delta}</math>
where {{math|1=''M'' = ''δ''(cos ''δ'')<sup>−Re ''s''</sup> ''e''<sup>Im ''sδ''</sup>}} is a constant independent of {{mvar|u}} or {{mvar|R}}. Again referring to the behavior of {{math|''x''<sup>''n''</sup> ''e''<sup>−''x''</sup>}} for large {{mvar|x}}, we see that the last expression approaches 0 as {{mvar|R}} increases towards {{math|∞}}.
In total we now have:
<math display="block">\Gamma(s) = \lim_{|z| \to \infty} \gamma(s, z), \quad \left|\arg z\right| < \pi/2 - \epsilon,</math>
if {{mvar|s}} is not a non-negative integer, {{math|0 < ''ε'' < ''π''/2}} is arbitrarily small, but fixed, and {{math|''γ''}} denotes the principal branch on this domain.

=====Overview=====
<math>\gamma(s, z)</math> is:
* [[Entire function|entire]] in {{mvar|z}} for fixed, positive integer {{mvar|s}};
* multi-valued [[Holomorphic function|holomorphic]] in {{mvar|z}} for fixed {{mvar|s}} not an integer, with a [[branch point]] at {{math|1=''z'' = 0}};
* on each branch [[meromorphic]] in {{mvar|s}} for fixed {{math|1=''z'' ≠ 0}}, with simple poles at non-positive integers s.