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| | You start in a pair of a lovely island where your amazing peaceful village is in beaches and woods right up until the enemies known just like the BlackGuard led by Lieutenant Hammerman invades your destination. After managing to guard against a little invasion force, he provides avenge his loss while battle.<br><br> |
| {{Refimprove|date=August 2009}}
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| {{Original research|date=August 2009}}
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| In [[mathematics]], the upper and the lower ''incomplete gamma functions'' are respectively as follows:
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| :<math> \Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t ,\,\! \qquad \gamma(s,x) = \int_0^x t^{s-1}\,e^{-t}\,{\rm d}t .\,\!</math>
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| ==Properties==
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| In both cases ''s'' is a complex parameter, such that the real part of ''s'' is positive.
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| By [[integration by parts]] we find the ''recurrence relations''
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| :<math>\Gamma(s,x)= (s-1)\Gamma(s-1,x) + x^{s-1} e^{-x}</math>
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| and conversely
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| :<math> \gamma(s,x) =(s-1)\gamma(s-1,x) - x^{s-1} e^{-x}</math>
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| Since the ordinary gamma function is defined as
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| :<math> \Gamma(s) = \int_0^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t</math>
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| we have
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| :<math> \Gamma(s) = \Gamma(s,0)</math>
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| and
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| :<math> \gamma(s,x) + \Gamma(s,x) = \Gamma(s).</math>
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| ===Continuation to complex values===
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| The lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive ''s'' and ''x'', can be developed into [[holomorphic function]]s, with respect both to ''x'' and ''s'', defined for almost all combinations of complex x and s.<ref>[http://dlmf.nist.gov/8.2.ii DLMF, Incomplete Gamma functions, analytic continuation]</ref> Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.
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| ====Lower incomplete Gamma function====
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| =====Holomorphic extension=====
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| Repeated application of the recurrence relation for the '''lower incomplete gamma''' function leads to the [[power series]] expansion: [http://dlmf.nist.gov/8.8.E7]
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| :<math>\gamma(s, x) = \sum_{k=0}^\infty \frac{x^s e^{-x} x^k}{s(s+1)...(s+k)} = x^s \, \Gamma(s) \, e^{-x}\sum_{k=0}^\infty\frac{x^k}{\Gamma(s+k+1)}</math>
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| Given the [[Gamma function#Approximations|rapid growth]] in [[Absolute value#Complex numbers|absolute value]] of [[Gamma function|Γ(''z'' + ''k'')]] when ''k'' → ∞, and the fact that the [[Reciprocal Gamma function|reciprocal of 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 ''s'' and ''x''. By a theorem of Weierstraß,<ref>[http://www.math.washington.edu/~marshall/math_534/Notes.pdf] Theorem 3.9 on p.56</ref> the limiting function, sometimes denoted as <math>\gamma^*</math>,
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| :<math>\gamma^*(s, z) := e^{-z}\sum_{k=0}^\infty\frac{z^k}{\Gamma(s+k+1)}</math> [http://dlmf.nist.gov/8.7.E1]
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| is [[Entire function|entire]] with respect to both ''z'' (for fixed ''s'') and ''s'' (for fixed ''z'') [http://dlmf.nist.gov/8.2.ii], and, thus, holomorphic on ℂ×ℂ by [[Hartog's theorem]][http://www.math.umn.edu/~garrett/m/complex/hartogs.pdf]. Hence, the following ''decomposition''
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| :<math>\gamma(s,z) = z^s \, \Gamma(s) \, \gamma^*(s,z)</math> [http://dlmf.nist.gov/8.2.E6],
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| extends the real lower incomplete gamma function as a [[Holomorphic function|holomorphic]] function, both jointly and separately in ''z'' and ''s''. It follows from the properties of z<sup>s</sup> and the [[Gamma function|Γ-function]], that the first two factors capture the [[Mathematical singularity|singularities]] of γ (at ''z'' = 0 or ''s'' a non-positive integer), whereas the last factor contributes to its zeros.
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| =====Multi-valuedness=====
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| The [[complex logarithm]] log ''z'' = log |''z''| + ''i'' arg ''z'' is determined up to a multiple of 2πi only, which renders it [[Multi-valued function|multi-valued]]. Functions involving the complex logarithm typically inherit this property. Among these are the [[Exponentiation#Powers of complex numbers|complex power]], and, since ''z''<sup>''s''</sup> appears in its decomposition, the γ-function, too.
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| The indeterminacy of multi-valued functions introduces complications, since it must be stated how to select a value. Strategies to handle this are:
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| * (the most general way) replace the domain ℂ of multi-valued functions by a suitable manifold in ℂ×ℂ called [[Riemann surface]]. While this removes multi-valuedness, one has to know the theory behind it [http://math.berkeley.edu/~teleman/math/Riemann.pdf];
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| * restrict the domain such that a multi-valued function decomposes into separate single-valued [[Branch point|branches]], which can be handled individually.
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| The following set of rules can be used to interpret formulas in this section correctly. If not mentioned otherwise, the following is assumed:
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| ======Sectors======
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| Sectors in ℂ having their vertex at ''z'' = 0 often prove to be appropriate domains for complex expressions. A sector D consists of all complex ''z'' fulfilling ''z'' ≠ 0 and ''α'' − ''δ'' < arg ''z'' < ''α'' + ''δ'' with some ''α'' and 0 < ''δ'' ≤ ''π''. Often, ''α'' can be arbitrarily chosen and is not specified then. If ''δ'' is not given, it is assumed to be π, and the sector is in fact the whole plane ℂ, with the exception of a half-line originating at ''z'' = 0 and pointing into the direction of −''α'', usually serving as a [[Branch cut#Branch cuts|branch cut]]. Note: In many applications and texts, ''α'' is silently taken to be 0, which centers the sector around the positive real axis.
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| ======Branches======
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| In particular, a single-valued and holomorphic logarithm exists on any such sector D having its imaginary part bound to the range (''α'' − ''δ'', ''α'' + ''δ''). Based on such a restricted logarithm, ''z''<sup>''s''</sup> and the incomplete gamma functions in turn collapse to single-valued, holomorphic functions on ''D'' (or ''ℂ''×''D''), called branches of their multi-valued counterparts on D. Adding a multiple of 2π to ''α'' yields a different set of correlated branches on the same set ''D''. However, in any given context here, ''α'' is assumed fixed and all branches involved are associated to it. If |''α''| < ''δ'', the branches are called [[principal branch|principal]], because they equal their real analogons on the positive real axis. Note: In many applications and texts, formulas hold only for principal branches.
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| ======The exponential function ''e''<sup>''s''</sup>======
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| The expression ''e''<sup>''s''</sup> shall always denote the [[Exponential function#Complex plane|exponential function]], which is the restriction of a principal branch of ''z''<sup>''s''</sup> to ''z'' = ''e''.
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| ======Relation between branches======
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| 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 by <math>e^{s*2k\pi i}</math>[http://dlmf.nist.gov/8.2.E8], ''k'' a suitable integer.
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| =====Behavior near branch point=====
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| The decomposition above further shows, that γ behaves near ''z'' = 0 [[asymptotic]]ally like:
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| :<math>\gamma(s, z) \asymp z^s \, \Gamma(s) \, \gamma^*(s, 0) = z^s \, \Gamma(s)/\Gamma(s+1) = z^s/s</math>
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| For positive real ''x'', ''y'' and ''s'', ''x''<sup>''y''</sup>/y → 0, when (''x'', ''y'') → (0, ''s''. This seems to justify setting ''γ(s, 0) = 0'' for real ''s'' > 0. However, matters are somewhat different in the complex realm. Only if (a) the real part of ''s'' is positive, and (b) values ''u''<sup>''v''</sup> are taken from just a finite set of branches, they are guaranteed to converge to zero as (''u'', ''v'') → (0, ''s''), and so does ''γ''(''u'', ''v''). On a single [[branch point|branch]] of ''γ''(''b'') is naturally fulfilled, so '''there''' ''γ''(''s'', 0) = 0 for ''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]].
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| =====Algebraic relations=====
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| All algebraic relations and differential equations observed by the real ''γ''(''s'', ''z'') hold for its holomorphic counterpart as well. This is a consequence of the identity theorem [http://planetmath.org/encyclopedia/RigidityTheoremForAnalyticFunctions.html], stating that equations between holomorphic functions valid on a real interval, hold everywhere. In particular, the recurrence relation [http://dlmf.nist.gov/8.8.E1] and ''∂γ''(''s'',''z'')/''∂z'' = ''z''<sup>''s''−1</sup> ''e''<sup>−''z''</sup> [http://dlmf.nist.gov/8.8.E12] are preserved on corresponding branches.
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| =====Integral representation=====
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| The last relation tells us, that, for fixed ''s'', ''γ'' is a [[Primitive function|primitive or antiderivative]] of the holomorphic function ''z''<sup>''s''−1</sup> ''e''<sup>−''z''</sup>. Consequently [http://planetmath.org/encyclopedia/ComplexAntiderivative.html], for any complex ''u'', ''v'' ≠ 0,
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| :<math>\int_u^v t^{s-1}\,e^{-t}\,{\rm d}t = \gamma(s,v) - \gamma(s,u)</math>
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| 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 ''s'' is positive, then the limit ''γ''(''s'', ''u'') → 0 for ''u'' → 0 applies, finally arriving at the complex integral definition of ''γ''
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| :<math>\gamma(s, z) = \int_0^z t^{s-1}\,e^{-t}\,{\rm d}t, \, \Re(s) > 0. </math>[http://dlmf.nist.gov/8.2.E1]
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| 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 0 and ''z''.
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| =====Limit for ''z'' → +∞=====
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| ======Real values======
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| Given the integral representation of a principal branch of γ, the following equation holds for all positive real s, x:[http://dlmf.nist.gov/5.2.E1]
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| :<math>\Gamma(s) = \int_0^\infty t^{s-1}\,e^{-t}\,{\rm d}t = \lim_{x \rightarrow \infty} \gamma(s, x)</math>
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| ======''s'' complex======
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| This result extends to complex s. Assume first {{math|1 ≤ Re(s) ≤ 2}} and {{math|1 < a < b}}. Then
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| :<math>|\gamma(s, b) - \gamma(s, a)| \le \int_a^b |t^{s-1}| e^{-t}\,{\rm d}t = \int_a^b t^{\Re s-1} e^{-t}\,{\rm d}t \le \int_a^b t e^{-t}\,{\rm d}t</math>
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| where
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| :<math>|z^s| = |z|^{\Re s}\,e^{-\Im s\arg z}</math>[http://dlmf.nist.gov/4.4.E15]
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| has been used in the middle. Since the final integral becomes arbitrarily small if only ''a'' is large enough, γ(s, x) converges uniformly for ''x'' → ∞ on the strip {{math|1 ≤ Re(s) ≤ 2}} towards a holomorphic function,<ref>[http://www.math.washington.edu/~marshall/math_534/Notes.pdf] Theorem 3.9 on p.56</ref> which must be Γ(s) because of the identity theorem [http://planetmath.org/encyclopedia/RigidityTheoremForAnalyticFunctions.html]. Taking the limit in the recurrence relation ''γ''(''s'',''x'') = (''s'' − 1)''γ''(''s'' − 1,''x'') − ''x''<sup>s''
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| −1</sup> ''e''<sup>−''x''</sup> and noting, that lim ''x''<sup>''n''</sup> ''e''<sup>−''x''</sup> = 0 for ''x'' → ∞ and all n, shows, that γ(s,x) converges outside the strip, too, towards a function obeying the recurrence relation of the Γ-function. It follows
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| :<math>\Gamma(s) = \lim_{x \rightarrow \infty} \gamma(s, x)</math>
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| for all complex ''s'' not a non-positive integer, x real and γ principal.
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| ======Sectorwise convergence======
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| Now let ''u'' be from the sector |arg ''z''| < ''δ'' < ''π''/2 with some fixed ''δ'' (''α'' = 0), ''γ'' be the principal branch on this sector, and look at
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| :<math>\Gamma(s) - \gamma(s, u) = \Gamma(s) - \gamma(s, |u|) + \gamma(s, |u|) - \gamma(s, u).</math>
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| As shown above, the first difference can be made arbitrarily small, if |''u''| is sufficiently large. The second difference allows for following estimation:
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| :<math>|\gamma(s, |u|) - \gamma(s, u)| \le \int_u^{|u|} |z^{s-1} e^{-z}|\,{\rm d}z = \int_u^{|u|} |z|^{\Re s - 1}\,e^{-\Im s\,\arg z}\,e^{-\Re z} \,{\rm d}z</math>
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| where we made use of the integral representation of γ and the formula about |z<sup>s</sup>| above. If we integrate along the arc with radius ''R'' = |''u''| around 0 connecting ''u'' and |''u''|, then the last integral is
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| :<math>\le R|\arg u|\,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>
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| where ''M'' = ''δ''(cos ''δ'')<sup>−Re ''s''</sup> ''e''<sup>Im ''sδ''</sup> is a constant independent of ''u'' or ''R''. Again referring to the behavior of ''x''<sup>''n''</sup> ''e''<sup>−''x''</sup> for large ''x'', we see that the last expression approaches 0 as ''R'' increases towards ∞.
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| In total we now have:
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| :<math>\Gamma(s) = \lim_{|z| \rightarrow \infty} \gamma(s, z), \quad |\arg z| < \pi/2 - \epsilon</math>
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| if ''s'' is not a non-negative integer, 0 < ''ε'' < ''π''/2 is arbitrarily small, but fixed, and ''γ'' denotes the principal branch on this domain.
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| =====Overview=====
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| <math>\gamma(s, z)</math> is:
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| * [[Entire function|entire]] in ''z'' for fixed, positive integral s;
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| * multi-valued [[Holomorphic function|holomorphic]] in ''z'' for fixed ''s'' not an integer, with a [[branch point]] at ''z'' = 0;
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| * on each branch [[meromorphic]] in ''s'' for fixed ''z'' ≠ 0, with simple poles at non-positive integers s.
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| ====Upper incomplete Gamma function====
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| As for the '''upper incomplete gamma function''', a [[Holomorphic function|holomorphic]] extension, with respect to ''z'' or ''s'', is given by
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| : <math>\Gamma(s,z) = \Gamma(s) - \gamma(s, z)</math> [http://dlmf.nist.gov/8.2.E3]
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| at points (''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>.
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| When ''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 ''s'' → 0, fills in the missing values. [[Complex analysis]] guarantees [[holomorphy|holomorphicity]], because <math>\Gamma(s,z)</math> proves to be [[Bounded function|bounded]] in a [[Neighbourhood (mathematics)|neighbourhood]] of that limit for a fixed ''z''[http://planetmath.org/encyclopedia/RiemannsRemovableSingularityTheorem.html].
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| To determine the limit, the power series of <math>\gamma^*</math> at ''z'' = 0 turns out useful. When replacing <math>e^{-x}</math> by its power series in the integral definition of <math>\gamma</math>, one obtains (assume ''x'',''s'' positive reals for now):
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| :<math>\gamma(s, x) = \int_0^x t^{s-1} e^{-t} \operatorname{d}t = \int_0^x \sum_{k=0}^\infty (-1)^k\,\frac{t^{s+k-1}}{k!}\operatorname{d}t = \sum_{k=0}^\infty (-1)^k\,\frac{x^{s+k}}{k!(s+k)} = x^s\,\sum_{k=0}^\infty \frac{(-x)^k}{k!(s+k)}</math>
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| or
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| :<math>\gamma^*(s,x) = \sum_{k=0}^\infty \frac{(-x)^k}{k!\,\Gamma(s)(s+k)}.</math> [http://dlmf.nist.gov/8.7.E1]
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| which, as a series representation of the entire <math>\gamma^*</math> function, converges for all complex ''x'' (and all complex ''s'' not a non-positive integer).
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| With its restriction to real values lifted, the series allows the expansion:
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| :<math>\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{(-z)^k}{k!(s+k)},\quad \Re(s) > -1, \,s \ne 0</math>
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| When ''s'' → 0:
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| :<math>\frac{z^s-1}{s} \rightarrow \ln(z),\quad \Gamma(s) - \frac{1}{s} = \frac{1}{s} - \gamma + O(s) - \frac{1}{s} \rightarrow-\gamma</math>,<ref>[[Gamma function#General|see last eq.]]</ref>
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| (<math>\gamma</math> is the [[Euler–Mascheroni constant]] here), hence,
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| :<math>\Gamma(0,z) = \lim_{s\rightarrow 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>
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| is the limiting function to the upper incomplete gamma function as ''s'' → 0, also known as [[Exponential integral|<math>E_1(z)</math>]].<ref>http://dlmf.nist.gov/8.4.E4</ref>
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| By way of the recurrence relation, values of <math>\Gamma(-n, z)</math> for positive integers ''n'' can be derived from this result, so the upper incomplete gamma function proves to exist and be holomorphic, with respect both to ''z'' and ''s'', for all ''s'' and ''z'' ≠ 0.
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| <math>\Gamma(s, z)</math> is:
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| * [[Entire function|entire]] in ''z'' for fixed, positive integral s;
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| * multi-valued [[Holomorphic function|holomorphic]] in ''z'' for fixed ''s'' non zero and not a positive integer, with a [[branch point]] at ''z'' = 0;
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| * = <math>\Gamma(s)</math> for ''s'' with positive real part and ''z'' = 0 (the limit when <math>(s_i,z_i) \rightarrow (s, 0)</math>), but this is a continuous extension, not an [[analytic continuation|analytic one]] ('''does not''' hold for real s<0!);
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| * on each branch [[Entire function|entire]] in ''s'' for fixed ''z'' ≠ 0.
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| ===Special values===
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| * <math>\Gamma(s) = (s-1)!</math> if ''s'' is a positive [[integer]],
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| * <math>\Gamma(s,x) = (s-1)!\, e^{-x} \sum_{k=0}^{s-1} \frac{x^k}{k!}</math> if ''s'' is a positive [[integer]],<ref>{{mathworld|urlname=IncompleteGammaFunction | title=Incomplete Gamma Function}} (equation 2)</ref>
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| * <math> \Gamma(s,0) = \Gamma(s), \Re(s) > 0</math>
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| * <math>\Gamma(1,x) = e^{-x},</math>
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| * <math>\gamma(1,x) = 1 - e^{-x},</math>
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| * <math>\Gamma(0,x) = -{\rm Ei}(-x)</math> for <math>x>0,</math>
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| * <math>\Gamma(s,x) = x^s \, {\rm E}_{1-s}(x),</math>
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| * <math>\Gamma\left(\tfrac{1}{2}, x\right) = \sqrt\pi\,{\rm erfc}\left(\sqrt x\right),</math>
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| * <math>\gamma\left(\tfrac{1}{2}, x\right) = \sqrt\pi\,{\rm erf}\left(\sqrt x\right).</math>
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| Here, <math>\mathrm{Ei}</math> is the [[exponential integral]], <math>\mathrm{E_n}</math> is the [[Exponential integral#Relation with other functions|generalized exponential integral]], <math>\mathrm{erf}</math> is the [[error function]], and <math>\mathrm{erfc}</math> is the [[complementary error function]], <math>\operatorname{erfc}(x) = 1-\operatorname{erf}(x)</math>.
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| ===Asymptotic behavior===
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| * <math> \frac{\gamma(s,x)}{x^s} \rightarrow \frac 1 s</math> as <math>x \rightarrow 0,</math>
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| * <math> \frac{\Gamma(s,x)}{x^s} \rightarrow -\frac 1 s</math> as <math>x \rightarrow 0</math> and <math>\Re (s) < 0\,</math>
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| * <math> \gamma(s,x) \rightarrow \Gamma(s)</math> as <math>x \rightarrow \infty,</math>
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| * <math> \frac{\Gamma(s,x)}{x^{s-1} e^{-x}} \rightarrow 1</math> as <math>x \rightarrow \infty,</math>
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| * <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>|\!\arg z| < \tfrac{3}{2} \pi</math>.<ref>[http://dlmf.nist.gov/8.11.i DLMF, Incomplete Gamma functions, 8.11(i)]</ref>
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| ==Evaluation formulae==
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| The lower gamma function has the straight forward expansion
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| :<math>\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>
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| where ''M'' is Kummer's [[confluent hypergeometric function]].
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| ===Connection with Kummer's confluent hypergeometric function===
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| When the real part of ''z'' is positive,
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| :<math>
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| \gamma(s,z) = \frac{}{} s^{-1} z^s e^{-z} M(1,s+1,z)
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| </math>
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| where
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| :<math>
| |
| 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>
| |
| \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)}{\rm d}u =
| |
| \\
| |
| &= e^{-z} z^s U(1,1+s,z) = e^{-z} \int_0^\infty e^{-u} (z+u)^{s-1}{\rm d} u = e^{-z} z^s \int_0^\infty e^{-z u} (1+u)^{s-1}{\rm d} u.
| |
| \end{align}
| |
| </math>
| |
| | |
| For the actual computation of numerical values, [[Gauss's continued fraction]] provides a useful expansion:
| |
| | |
| :<math>
| |
| \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 ''z'', provided only that ''s'' is not a negative integer.
| |
| | |
| The upper gamma function has the continued fraction
| |
| :<math>
| |
| \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><ref>Abramowitz and Stegun [http://www.math.sfu.ca/~cbm/aands/page_263.htm p. 263, 6.5.31]</ref>
| |
| and
| |
| :<math>
| |
| \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>{{Citation needed|date=February 2013}}
| |
| | |
| ===Multiplication theorem===
| |
| The following [[multiplication theorem]] holds true:
| |
| :<math>
| |
| \begin{align}
| |
| \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).
| |
| \end{align}
| |
| </math>
| |
| | |
| == Regularized Gamma functions and Poisson random variables ==
| |
| | |
| Two related functions are the regularized Gamma functions:
| |
| | |
| :<math>P(s,x)=\frac{\gamma(s,x)}{\Gamma(s)},</math>
| |
| | |
| :<math>Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}=1-P(s,x).</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>0</math> is an integer, <math>Q(s,\lambda)</math> is the cumulative distribution function for [[Poisson random variable]]s: If <math>X</math> is a <math>{\rm Poi}(\lambda)</math> random variable then
| |
| | |
| :<math> Pr(X<s) = \sum_{i<s} e^{-\lambda} \frac{\lambda^i}{i!} = \frac{\Gamma(s,\lambda)}{\Gamma(s)} = Q(s,\lambda).
| |
| </math>
| |
| | |
| This formula can be derived by repeated integration by parts.
| |
| | |
| == Derivatives ==
| |
| | |
| The derivative of the upper incomplete gamma function <math> \Gamma (s,x) </math> with respect to ''x'' is well known. It is simply given by the integrand of its integral definition:
| |
| :<math>
| |
| \frac{\partial \Gamma (s,x) }{\partial x} = - \frac{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. 149-165, [http://www.springerlink.com/content/t7571u653t83037j/]
| |
| </ref>
| |
| :<math>
| |
| \frac{\partial \Gamma (s,x) }{\partial s} = \ln x \Gamma (s,x) + x\,T(3,s,x)
| |
| </math>
| |
| and the second derivative by
| |
| :<math>
| |
| \frac{\partial^2 \Gamma (s,x) }{\partial s^2} = \ln^2 x \Gamma (s,x) + 2 x[\ln x\,T(3,s,x) + T(4,s,x) ]
| |
| </math>
| |
| where the function <math>T(m,s,x)</math> is a special case of the [[Meijer G-function]]
| |
| :<math>
| |
| 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>
| |
| \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>
| |
| P_j^n = \left( \begin{array}{l} n \\ j \end{array} \right) j! = \frac{n!}{(n-j)!}.
| |
| </math>
| |
| All such derivatives can be generated in succession from:
| |
| :<math>
| |
| \frac{\partial T (m,s,x) }{\partial s} = \ln x ~ T(m,s,x) + (m-1) T(m+1,s,x)
| |
| </math>
| |
| and
| |
| :<math>
| |
| \frac{\partial T (m,s,x) }{\partial x} = -\frac{1}{x} [T(m-1,s,x) + T(m,s,x)]
| |
| </math>
| |
| This function <math>T(m,s,x)</math> can be computed from its series representation valid for <math> |z| < 1 </math>,
| |
| :<math>
| |
| T(m,s,z) = - \frac{(-1)^{m-1} }{(m-2)! } \frac{{\rm d}^{m-2} }{{\rm d}t^{m-2} } \left[\Gamma (s-t) z^{t-1}\right]\Big|_{t=0} + \sum_{n=0}^{\infty} \frac{(-1)^n z^{s-1+n}}{n! (-s-n)^{m-1} }
| |
| </math>
| |
| with the understanding that ''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) = {\rm E}_1(x)</math>, where <math>{\rm 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. Milgram|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|ref=harv}}</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>
| |
| \int_{x}^{\infty} \frac{t^{s-1} \ln^m t}{e^t} {\rm d}t= \frac{\partial^m}{\partial s^m} \int_{x}^{\infty} \frac{t^{s-1}}{e^t} {\rm d}t= \frac{\partial^m}{\partial s^m} \Gamma (s,x)
| |
| </math>
| |
| This formula can be further ''inflated'' or generalized to a huge class of [[Laplace transforms]] 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]]:
| |
| | |
| :<math>
| |
| \int x^{b-1} \gamma(s,x) \mathrm d x = \frac{1}{b} \left( x^b \gamma(s,x) + \Gamma(s+b,x) \right).
| |
| </math>
| |
| | |
| :<math>
| |
| \int x^{b-1} \Gamma(s,x) \mathrm d x = \frac{1}{b} \left( x^b \Gamma(s,x) - \Gamma(s+b,x) \right),
| |
| </math>
| |
| | |
| The lower and the upper incomplete Gamma function are connected via the [[Fourier transform]]:
| |
| | |
| :<math>
| |
| \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} \mathrm d z = \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|1980|loc= § 7.642}}.
| |
| | |
| == Notes ==
| |
| <references/>
| |
| | |
| == References ==
| |
| * {{AS ref|6.5}} {{Cite web | title= Incomplete Gamma function | url= http://www.math.sfu.ca/~cbm/aands/page_260.htm}} §6.5.
| |
| * {{Cite journal| first1=Giampietro | last1=Allasia
| |
| |first2=Renata |last2=Besenghi
| |
| |title= Numerical calculation of incomplete gamma functions by the trapezoidal rule
| |
| |journal=Numer. Math.
| |
| |volume=50
| |
| |issue=4
| |
| |doi=10.1007/BF0139666
| |
| |year=1986
| |
| |pages=419–428
| |
| }}
| |
| * {{cite journal|first1=Paolo |last1=Amore
| |
| |title=Asymptotic and exact series representations for the incomplete Gamma function
| |
| |journal=Europhys. Lett.
| |
| |year=2005
| |
| |mr=2170316
| |
| |volume=71 | issue=1
| |
| |pages=1–7
| |
| }}
| |
| * G. Arfken and H. Weber. ''Mathematical Methods for Physicists''. Harcourt/Academic Press, 2000. ''(See Chapter 10.)''
| |
| * {{Cite journal | last1= DiDonato | first1= Armido R. | last2= Morris, Jr. | first2= Alfred H. | title= Computation of the incomplete gamma function ratios and their inverse | journal= ACM Transactions on Mathematical Software (TOMS) | volume= 12 | issue= 4 | pages= 377–393 |date=Dec 1986 | doi= 10.1145/22721.23109 | ref= harv}}
| |
| * {{cite journal|first1= Richard |last1=Barakat
| |
| |title=Evaluation of the Incomplete Gamma Function of Imaginary Argument by Chebyshev Polynomials
| |
| |journal=Math. Comp.
| |
| |year=1961
| |
| |mr=0128058
| |
| |volume=15 | issue=73 |pages=7–11
| |
| }}
| |
| * {{cite journal|first1=Petr |last1=Carsky |first1=Martin |last2=Polasek
| |
| |title=Incomplete Gamma {{math|F_m(x)}} functions for real and complex arguments
| |
| |year=1998 |doi =10.1006/jcph.1998.5975 |volume=143
| |
| |issue=1 |pages=259–265 |journal=J. Comput. Phys. | mr=1624704
| |
| }}
| |
| * {{cite journal|first1= M. Aslam |last1=Chaudhry |first2=S. M. |last2=Zubair
| |
| |title=On the decomposition of generalized incomplete Gamma functions with applications to Fourier transforms
| |
| |journal=J. Comput. Appl. Math.
| |
| |year=1995 |volume=59 |issue=101 | pages=253–284 | mr=1346414
| |
| }}
| |
| * {{Cite journal | last1= DiDonato | first1= Armido R. | last2= Morris, Jr. | first2= Alfred H. | doi= 10.1145/29380.214348 | title= ALGORITHM 654: FORTRAN subroutines for computing the incomplete gamma function ratios and their inverse | journal= ACM Transactions on Mathematical Software (TOMS) | volume= 13 | issue= 3 | pages= 318–319 |date=Sep 1987 | ref= harv}} ''(See also [http://www.netlib.org/toms/654 www.netlib.org/toms/654]).''
| |
| * {{Cite journal|first1=H. | last1=Früchtl | first2=P. |last2=Otto
| |
| |title= A new algorithm for the evaluation of the incomplete Gamma Function on vector computers
| |
| |journal= ACM Trans. Math. Softw.
| |
| |year=1994 |volume=20 |issue=4|pages=436–446
| |
| }}
| |
| * {{cite journal|first1=Walter |last1=Gautschi
| |
| |title=The incomplete gamma function since Tricomi
| |
| |year=1998 | journal=Atti Convegni Lincei | mr=1737497
| |
| |volume=147 | pages=203–237
| |
| }}
| |
| * {{cite journal|first1=Walter |last1=Gautschi
| |
| |title = A Note on the recursive calculation of Incomplete Gamma Functions
| |
| |journal=ACM Trans. Math. Softw.
| |
| |year=1999
| |
| |mr=1697463 |volume=25 |issue=1 | pages=101–107
| |
| }}
| |
| * {{cite book | last1= Gradshteyn | first1= I.S. | last2= Ryzhik | first2= I.M. | title= Tables of Integrals, Series, and Products | edition= 4th | location= New York | publisher= Academic Press | year= 1980 | isbn= 0-12-294760-6 | ref= harv}} ''(See Chapter 8.35.)''
| |
| * {{cite journal|first1=William B. |last1=Jones | first2= W. J. |last2=Thron
| |
| |title=On the computation of incomplete gamma functions in the complex domain
| |
| |year=1985
| |
| |journal=J. Comp. Appl. Math | doi=10.1016/0377-0427(85)90034-2
| |
| |mr=0793971 | volume=12-13 | pages=401–417
| |
| }}
| |
| * {{springer|title=Incomplete gamma-function|id=p/i050470}}
| |
| * {{Cite journal | last1= Mathar | first1= Richard J. | title= Numerical representation of the incomplete gamma function of complex-valued argument | journal= Numerical Algorithms | year= 2004 | volume= 36 | issue= 3 | pages= 247–264 | doi= 10.1023/B:NUMA.0000040063.91709.5 | mr= 2091195 | ref= harv }}
| |
| * {{cite journal| first1=Allen R. | last1=Miller | first2=Ira S. | last2=Moskowitz
| |
| |title=On certain Generalized incomplete Gamma functions
| |
| |year=1998 | journal=J. Comput. Appl. Math
| |
| |volume=91 | issue=2|pages=179–190
| |
| }}
| |
| * {{dlmf | id= 8 | Incomplete Gamma and Related Functions | last= Paris | first= R. B.}}
| |
| * {{cite journal|first1=R. B. | last1=Paris |
| |
| title=A uniform asymptotic expansion for the incomplete gamma function
| |
| |journal= J. Comput. Appl. Math.
| |
| |year=2002 | doi=10.1016/S0377-0427(02)00553-8
| |
| |volume=148 | issue=2 | pages=323–339 | mr=1936142
| |
| }}
| |
| * {{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | publication-place=New York | isbn=978-0-521-88068-8 | chapter=Section 6.2. Incomplete Gamma Function and Error Function | chapter-url=http://apps.nrbook.com/empanel/index.html?pg=259 | ref=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
| |
| * {{cite journal|first1=Roy | last1=Takenaga | title=On the Evaluation of the Incomplete Gamma Function
| |
| |year=1966 | journal=Math. Comp. | volume=20 | issue=96
| |
| |pages=606–610|mr=0203911 |doi=10.1090/S0025-5718-1966-0203911-3
| |
| }}
| |
| * {{cite journal|first1=Nico | last1=Temme
| |
| |title=Uniform Asymptotic Expansions of the Incomplete Gamma Functions and the Incomplete Beta Function
| |
| |journal=Math. Comp.
| |
| |doi=10.1090/S0025-5718-1975-0387674-2
| |
| |volume=29 | issue=132 | year=1975 | pages=1109–1114 |mr=0387674
| |
| }}
| |
| * {{cite journal|first1=Riho | last1=Terras
| |
| |title=The determination of incomplete Gamma Functions through analytic integration
| |
| |year=1979 | journal = J. Comp. Phys. | volume=31 | pages=146–151|mr=0531128
| |
| }}
| |
| * {{cite journal| first1=Francesco G. | last1=Tricomi
| |
| |title=Sulla funzione gamma incompleta
| |
| |year=1950|doi=10.1007/BF02428264 | journal=Ann. Mat. Pura Applic.
| |
| |volume=31|pages=263–279|mr=0047834
| |
| }}
| |
| * {{cite journal|first1=F. G. |last1=Tricomi
| |
| |title=Asymptotische Eigenschaften der unvollst. Gammafunktion|
| |
| year=1950|mr=0045253|journal=Math. Zeitsch. | pages=136–148
| |
| |volume=53|number=2
| |
| }}
| |
| * {{cite journal|first1=Joris | last1=van Deun | first2=Ronald | last2=Cools
| |
| |title=A stable recurrence for the incomplete gamma function with imaginary second argument
| |
| |journal=Numer. Math. |year=2006 | doi=10.1007/s00211-006-0026-1 | pages=445–456 | volume=104|mr=2249673
| |
| }}
| |
| * {{cite journal|first1=Serge | last1=Winitzki | title=Computing the incomplete gamma function to arbitrary precision
| |
| |year=2003 | journal=Lect. Not. Comp. Sci. | volume=2667 | pages=790–798 |mr=2110953}}
| |
| * {{MathWorld|title=Incomplete Gamma Function|id=IncompleteGammaFunction}}
| |
| | |
| ==Miscellaneous utilities==
| |
| * <math>P(a,x)</math> — [http://www.danielsoper.com/statcalc/calc33.aspx Incomplete Gamma Function Calculator]
| |
| * <math>Q(a,x)</math> — [http://www.danielsoper.com/statcalc/calc34.aspx Incomplete Gamma Function Calculator – Complemented]
| |
| * <math>\gamma(a,x)</math> — [http://www.danielsoper.com/statcalc/calc24.aspx Incomplete Gamma Function Calculator – Lower Limit of Integration]
| |
| * <math>\Gamma(a,x)</math> — [http://www.danielsoper.com/statcalc/calc23.aspx Incomplete Gamma Function Calculator – Upper Limit of Integration]
| |
| * [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]]
| |