Editing Incomplete gamma function

{{Short description|Types of special mathematical functions}}
{{Use dmy dates|date=December 2023}}
[[File:Upper incomplete gamma function.jpg|thumb|300x300px|The upper incomplete gamma function for some values of s: 0 (blue), 1 (red), 2 (green), 3 (orange), 4 (purple).]]
[[File:Plot of the regularized incomplete gamma function Q(2,z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the regularized incomplete gamma function Q(2,z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the regularized incomplete gamma function Q(2,z) in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D]]
In [[mathematics]], the '''upper''' and '''lower incomplete gamma functions''' are types of [[special functions]] which arise as solutions to various mathematical problems such as certain [[integral]]s.

Their respective names stem from their integral definitions, which are defined similarly to the [[gamma function]] but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity.

==Definition==
The upper incomplete gamma function is defined as:
<math display="block"> \Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\, dt ,</math>
whereas the lower incomplete gamma function is defined as:
<math display="block"> \gamma(s,x) = \int_0^x t^{s-1}\,e^{-t}\, dt .</math>
In both cases {{mvar|s}} is a complex parameter, such that the real part of {{mvar|s}} is positive.

==Properties==

By [[integration by parts]]  we find the [[recurrence relations]]
<math display="block"> \Gamma(s+1,x) = s\Gamma(s,x) + x^{s} e^{-x}</math>
and
<math display="block"> \gamma(s+1,x) = s\gamma(s,x) - x^{s} e^{-x}.</math>
Since the ordinary gamma function is defined as
<math display="block"> \Gamma(s) = \int_0^{\infty} t^{s-1}\,e^{-t}\, dt</math>
we have
<math display="block"> \Gamma(s) = \Gamma(s,0) = \lim_{x \to \infty} \gamma(s,x)</math>
and
<math display="block"> \gamma(s,x) + \Gamma(s,x) = \Gamma(s).</math>

===Continuation to complex values===

The lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive {{mvar|s}} and {{mvar|x}}, can be developed into [[holomorphic function]]s, with respect both to {{mvar|x}} and {{mvar|s}}, defined for almost all combinations of complex {{mvar|x}} and {{mvar|s}}.<ref name="auto3">{{Cite web|url=https://dlmf.nist.gov/8.2|title=DLMF: §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions | website = dlmf.nist.gov}}</ref> Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.

====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.

====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}}.

===Special values===

* <math>\Gamma(s+1,1) = \frac{\lfloor e s! \rfloor}{e} </math> if {{mvar|s}} is a positive [[integer]],
* <math>\Gamma(s,x) = (s-1)!\, e^{-x} \sum_{k=0}^{s-1} \frac{x^k}{k!}</math> if {{mvar|s}} is a positive [[integer]],<ref>{{mathworld | urlname=IncompleteGammaFunction | title=Incomplete Gamma Function}} (equation 2)</ref>
* <math> \Gamma(s,0) = \Gamma(s), \Re(s) > 0</math>,
* <math>\Gamma(1,x) = e^{-x}</math>,
* <math>\gamma(1,x) = 1 - e^{-x}</math>,
* <math>\Gamma(0,x) = -\operatorname{Ei}(-x)</math> for <math>x > 0</math>,
* <math>\Gamma(s,x) = x^s \operatorname{E}_{1-s}(x)</math>,
* <math>\Gamma\left(\tfrac{1}{2}, x\right) = \sqrt\pi \operatorname{erfc}\left(\sqrt x\right)</math>,
* <math>\gamma\left(\tfrac{1}{2}, x\right) = \sqrt\pi \operatorname{erf}\left(\sqrt x\right)</math>.

Here, <math>\operatorname{Ei}</math> is the [[exponential integral]], <math>\operatorname{E}_n</math> is the [[Exponential integral#Relation with other functions|generalized exponential integral]], <math>\operatorname{erf}</math> is the [[error function]], and <math>\operatorname{erfc}</math> is the [[complementary error function]], <math>\operatorname{erfc}(x) = 1 - \operatorname{erf}(x)</math>.

===Asymptotic behavior===

* <math>\frac{\gamma(s,x)}{x^s} \to \frac{1}{s}</math> as <math>x \to 0</math>,
* <math>\frac{\Gamma(s,x)}{x^s} \to -\frac{1}{s}</math> as <math>x \to 0</math> and <math>\Re (s) < 0</math> (for real {{math|''s''}}, the error of {{math|Γ(''s'', ''x'') ~ −''x''<sup>''s''</sup> / ''s''}} is on the order of {{math|''O''(''x''<sup>min{''s'' + 1, 0}</sup>)}} if {{math|''s'' ≠ −1}} and {{math|''O''(ln(''x''))}} if {{math|1=''s'' = −1}}),
* <math>\Gamma(s,x) \sim \Gamma(s) - \sum_{n=0}^\infty (-1)^n \frac{x^{s+n}}{n!(s+n)}</math> as an [[asymptotic series]] where <math>x\to0^+</math> and <math>s\neq 0,-1,-2,\dots</math>.<ref name="auto">{{cite book |last=Bender & Orszag |date=1978 |title = Advanced Mathematical Methods for Scientists and Engineers |publisher=Springer|bibcode=1978amms.book.....B }}</ref>
* <math>\Gamma(-N,x) \sim C_N + \frac{(-1)^{N+1}}{N!} \ln x - \sum_{n=0,n\ne N}^\infty (-1)^n \frac{x^{n-N}}{n!(n-N)}</math> as an [[asymptotic series]] where <math>x \to 0^+</math> and <math>N = 1, 2, \dots</math>, where <math display="inline">C_N = \frac{(-1)^{N+1}}{N!} \left( \gamma - \displaystyle\sum_{n=1}^N \frac{1}{n} \right)</math>, where <math>\gamma</math> is the [[Euler-Mascheroni constant]].<ref name="auto"/>
* <math>\gamma(s,x) \to \Gamma(s)</math> as <math>x \to \infty</math>,
* <math>\frac{\Gamma(s,x)}{x^{s-1} e^{-x}} \to 1</math> as <math>x \to \infty</math>,
* <math>\Gamma(s,z) \sim z^{s-1} e^{-z} \sum_{k=0} \frac {\Gamma(s)} {\Gamma(s-k)} z^{-k}</math> as an [[asymptotic series]] where <math>|z| \to \infty</math> and <math>\left|\arg z\right| < \tfrac{3}{2} \pi</math>.<ref>{{Cite web|url=https://dlmf.nist.gov/8.11|title=DLMF: §8.11 Asymptotic Approximations and Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions|website=dlmf.nist.gov}}</ref>

==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.

== Regularized gamma functions and Poisson random variables ==

Two related functions are the regularized gamma functions:
<math display="block">\begin{align}
P(s,x) &= \frac{\gamma(s,x)}{\Gamma(s)}, \\[1ex]
Q(s,x) &= \frac{\Gamma(s,x)}{\Gamma(s)} = 1 - P(s,x).
\end{align}</math>
<math>P(s,x)</math> is the [[cumulative distribution function]] for [[Gamma distribution|gamma random variables]] with [[shape parameter]] <math>s</math> and [[scale parameter]] 1.

When <math>s</math> is an integer, <math>Q(s+1, \lambda)</math> is the cumulative distribution function for [[Poisson random variable]]s: If <math>X</math> is a <math>\mathrm{Poi}(\lambda)</math> random variable then
<math display="block"> \Pr(X \leq s) = \sum_{i \leq s} e^{-\lambda} \frac{\lambda^i}{i!} = \frac{\Gamma(s+1,\lambda)}{\Gamma(s+1)} = Q(s+1,\lambda).</math>

This formula can be derived by repeated integration by parts.

In the context of the [[stable count distribution]], the <math> s </math> parameter can be regarded as inverse of Lévy's stability parameter <math> \alpha</math>: 
<math display="block"> Q(s,x) = \int_0^\infty e^{\left( -{x^s}/{\nu} \right)} \, \mathfrak{N}_{{1}/{s}}\left(\nu\right)  \, d\nu , \quad (s > 1)</math>
where <math>\mathfrak{N}_{\alpha}(\nu)</math> is a standard stable count distribution of shape <math> \alpha = 1/s < 1</math>.

<math>P(s,x)</math> and <math>Q(s, x)</math> are implemented as <code>gammainc</code><ref>{{Cite web|url=https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gammainc.html#scipy.special.gammainc|title=scipy.special.gammainc — SciPy v1.11.4 Manual|website=docs.scipy.org}}</ref> and <code>gammaincc</code><ref>{{Cite web|url=https://docs.scipy.org/doc/scipy/reference/generated/scipy.special.gammaincc.html|title=scipy.special.gammaincc — SciPy v1.11.4 Manual|website=docs.scipy.org}}</ref> in [[scipy]].

== Derivatives ==

Using the integral representation above, the derivative of the upper incomplete gamma function <math> \Gamma (s,x) </math> with respect to {{mvar|x}} is
<math display="block"> \frac{\partial \Gamma (s,x) }{\partial x} = - x^{s-1} e^{-x}</math>
The derivative with respect to its first argument <math>s</math> is given by<ref>[[Keith Geddes|K.O. Geddes]], M.L. Glasser, R.A. Moore and T.C. Scott, ''Evaluation of Classes of Definite Integrals Involving Elementary Functions via Differentiation of Special Functions'', AAECC (Applicable Algebra in Engineering, Communication and Computing), vol. 1, (1990), pp.&nbsp;149–165, [https://doi.org/10.1007%2FBF01810298] 
</ref>
<math display="block">\frac{\partial \Gamma (s,x) }{\partial s} = \ln x \Gamma (s,x) + x\,T(3,s,x)</math>
and the second derivative by
<math display="block">\frac{\partial^2 \Gamma (s,x) }{\partial s^2} = \ln^2 x \Gamma (s,x) + 2 x \left[\ln x\,T(3,s,x) + T(4,s,x) \right]</math>
where the function <math>T(m,s,x)</math> is a special case of the [[Meijer G-function]]
<math display="block">T(m,s,x) = G_{m-1,\,m}^{\,m,\,0} \!\left( \left. \begin{matrix} 0, 0, \dots, 0 \\ s-1, -1, \dots, -1 \end{matrix} \; \right| \, x \right).</math>
This particular special case has internal ''closure'' properties of its own because it can be used to express ''all'' successive derivatives. In general, 
<math display="block">\frac{\partial^m \Gamma (s,x) }{\partial s^m} = \ln^m x \Gamma (s,x) + m x\,\sum_{n=0}^{m-1} P_n^{m-1} \ln^{m-n-1} x\,T(3+n,s,x)</math>
where <math> P_j^n </math> is the [[permutation]] defined by the [[Pochhammer symbol]]:
<math display="block">P_j^n = \binom{n}{j} j! = \frac{n!}{(n-j)!}.</math>
All such derivatives can be generated in succession from:
<math display="block">\frac{\partial T (m,s,x) }{\partial s} = \ln x ~ T(m,s,x) + (m-1) T(m+1,s,x)</math>
and
<math display="block">\frac{\partial T (m,s,x) }{\partial x} = -\frac{T(m-1,s,x) + T(m,s,x)}{x}</math>
This function <math>T(m,s,x)</math> can be computed from its series representation valid for <math> |z| < 1 </math>, 
<math display="block">T(m,s,z) = - \frac{\left(-1\right)^{m-1} }{(m-2)! } \left.\frac{d^{m-2} }{dt^{m-2} } \left[\Gamma (s-t) z^{t-1}\right]\right|_{t=0} + \sum_{n=0}^{\infty} \frac{\left(-1\right)^n z^{s-1+n}}{n! \left(-s-n\right)^{m-1} }</math>
with the understanding that {{mvar|s}} is not a negative integer or zero.  In such a case, one must use a limit. Results for <math> |z| \ge 1 </math> can be obtained by [[analytic continuation]]. Some special cases of this function can be simplified. For example, <math>T(2,s,x)=\Gamma(s,x)/x</math>, <math>x\,T(3,1,x) = \mathrm{E}_1(x)</math>, where <math>\mathrm{E}_1(x)</math> is the [[Exponential integral]]. These derivatives and the function <math>T(m,s,x)</math> provide exact solutions to a number of integrals by repeated differentiation of the integral definition of the upper incomplete gamma function.<ref>{{cite journal | first1=M. S. |last1=Milgram|title=The generalized integro-exponential function|journal=Math. Comp.| year=1985|volume=44| issue=170| pages=443–458|mr=0777276| doi=10.1090/S0025-5718-1985-0777276-4|doi-access=free}}</ref><ref>{{cite arXiv| eprint=0912.3844| author1=Mathar|title=Numerical Evaluation of the Oscillatory Integral over exp(i*pi*x)*x^(1/x) between 1 and infinity| class=math.CA|year=2009}}, App B</ref>
For example,
<math display="block"> \int_{x}^{\infty} \frac{t^{s-1} \ln^m t}{e^t} dt= \frac{\partial^m}{\partial s^m} \int_{x}^{\infty} \frac{t^{s-1}}{e^t}  dt = \frac{\partial^m}{\partial s^m} \Gamma (s,x)</math>
This formula can be further ''inflated'' or generalized to a huge class of [[Laplace transform]]s and [[Mellin transform]]s. When combined with a [[computer algebra system]], the exploitation of special functions provides a powerful method for solving definite integrals, in particular those encountered by practical engineering applications (see [[Symbolic integration]] for more details).

==Indefinite and definite integrals==

The following indefinite integrals are readily obtained using [[integration by parts]] (with the [[constant of integration]] omitted in both cases):
<math display="block">\begin{align}
\int x^{b-1} \gamma(s,x) \, dx &= \frac{1}{b} \left( x^b \gamma(s,x) - \gamma(s+b,x) \right), \\[1ex]
\int x^{b-1} \Gamma(s,x) \, dx &= \frac{1}{b} \left( x^b \Gamma(s,x) - \Gamma(s+b,x) \right).
\end{align}</math>
The lower and the upper incomplete gamma function are connected via the [[Fourier transform]]:
<math display="block">\int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} dz = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}.</math>
This follows, for example, by suitable specialization of {{harv|Gradshteyn|Ryzhik|Geronimus|Tseytlin|2015|loc=§7.642}}<!-- location 7.642 does not match chapter given further below. Needs to be sorted out. -->.
== Notes ==
<references/>

== References ==
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* {{MathWorld|title=Incomplete Gamma Function|id=IncompleteGammaFunction}}
{{refend}}

== External links ==
* <math>P(a,x)</math> — [https://www.danielsoper.com/statcalc/calculator.aspx?id=33 Regularized Lower Incomplete Gamma Function Calculator]
* <math>Q(a,x)</math> — [https://www.danielsoper.com/statcalc/calculator.aspx?id=34 Regularized Upper Incomplete Gamma Function Calculator]
* <math>\gamma(a,x)</math> — [https://www.danielsoper.com/statcalc/calculator.aspx?id=24 Lower Incomplete Gamma Function Calculator]
* <math>\Gamma(a,x)</math> — [https://www.danielsoper.com/statcalc/calculator.aspx?id=23 Upper Incomplete Gamma Function Calculator]
* [http://functions.wolfram.com/GammaBetaErf/Gamma3/ formulas and identities of the Incomplete Gamma Function] functions.wolfram.com

{{DEFAULTSORT:Incomplete Gamma Function}}
[[Category:Gamma and related functions]]
[[Category:Continued fractions]]