Английская Википедия:Incomplete gamma function
Шаблон:Short description Шаблон:Use dmy dates
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 integrals.
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 Шаблон:Mvar is a complex parameter, such that the real part of Шаблон:Mvar 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 Шаблон:Mvar and Шаблон:Mvar, can be developed into holomorphic functions, with respect both to Шаблон:Mvar and Шаблон:Mvar, defined for almost all combinations of complex Шаблон:Mvar and Шаблон:Mvar.[1] 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: [2]
<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 of Шаблон:Math when Шаблон:Math, and the fact that the [[Reciprocal Gamma function|reciprocal of Шаблон:Math]] is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all complex Шаблон:Mvar and Шаблон:Mvar. By a theorem of Weierstraß,[3] the limiting function, sometimes denoted as <math>\gamma^*</math>,[4]
<math display="block">\gamma^*(s, z) := e^{-z}\sum_{k=0}^\infty\frac{z^k}{\Gamma(s+k+1)}</math>
is entire with respect to both Шаблон:Mvar (for fixed Шаблон:Mvar) and Шаблон:Mvar (for fixed Шаблон:Mvar),[1] and, thus, holomorphic on Шаблон:Math by Hartog's theorem.[5] Hence, the following decomposition
- <math>\gamma(s,z) = z^s \, \Gamma(s) \, \gamma^*(s,z),</math>[1]
extends the real lower incomplete gamma function as a holomorphic function, both jointly and separately in Шаблон:Mvar and Шаблон:Mvar. It follows from the properties of <math>z^s</math> and the Γ-function, that the first two factors capture the singularities of <math>\gamma(s,z)</math> (at Шаблон:Math or Шаблон:Mvar a non-positive integer), whereas the last factor contributes to its zeros.
Multi-valuedness
The complex logarithm Шаблон:Math is determined up to a multiple of Шаблон:Math only, which renders it multi-valued. Functions involving the complex logarithm typically inherit this property. Among these are the complex power, and, since Шаблон:Math appears in its decomposition, the Шаблон:Math-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 Шаблон:Math of multi-valued functions by a suitable manifold in Шаблон:Math called Riemann surface. While this removes multi-valuedness, one has to know the theory behind it;[6]
- restrict the domain such that a multi-valued function decomposes into separate single-valued 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 Шаблон:Math having their vertex at Шаблон:Math often prove to be appropriate domains for complex expressions. A sector Шаблон:Mvar consists of all complex Шаблон:Mvar fulfilling Шаблон:Math and Шаблон:Math with some Шаблон:Mvar and Шаблон:Math. Often, Шаблон:Mvar can be arbitrarily chosen and is not specified then. If Шаблон:Mvar is not given, it is assumed to be Шаблон:Pi, and the sector is in fact the whole plane Шаблон:Math, with the exception of a half-line originating at Шаблон:Math and pointing into the direction of Шаблон:Math, usually serving as a branch cut. Note: In many applications and texts, Шаблон:Mvar 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 Шаблон:Open-open. Based on such a restricted logarithm, Шаблон:Math and the incomplete gamma functions in turn collapse to single-valued, holomorphic functions on Шаблон:Mvar (or Шаблон:Math), called branches of their multi-valued counterparts on D. Adding a multiple of Шаблон:Math to Шаблон:Mvar yields a different set of correlated branches on the same set Шаблон:Mvar. However, in any given context here, Шаблон:Mvar is assumed fixed and all branches involved are associated to it. If Шаблон:Math, the branches are called 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>,[1] for Шаблон:Mvar a suitable integer.
Behavior near branch point
The decomposition above further shows, that γ behaves near Шаблон:Math asymptotically 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 Шаблон:Mvar, Шаблон:Mvar and Шаблон:Mvar, Шаблон:Math, when Шаблон:Math. This seems to justify setting Шаблон:Math for real Шаблон:Math. However, matters are somewhat different in the complex realm. Only if (a) the real part of Шаблон:Mvar is positive, and (b) values Шаблон:Math are taken from just a finite set of branches, they are guaranteed to converge to zero as Шаблон:Math, and so does Шаблон:Math. On a single branch of Шаблон:Math is naturally fulfilled, so there Шаблон:Math for Шаблон:Mvar with positive real part is a continuous limit. Also note that such a continuation is by no means an analytic one.
Algebraic relations
All algebraic relations and differential equations observed by the real Шаблон:Math 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 [2] and Шаблон:Math [2] are preserved on corresponding branches.
Integral representation
The last relation tells us, that, for fixed Шаблон:Mvar, Шаблон:Mvar is a primitive or antiderivative of the holomorphic function Шаблон:Math. Consequently, for any complex Шаблон:Math,
<math display="block">\int_u^v t^{s-1}\,e^{-t}\, dt = \gamma(s,v) - \gamma(s,u)</math>
holds, as long as the path of integration is entirely contained in the domain of a branch of the integrand. If, additionally, the real part of Шаблон:Mvar is positive, then the limit Шаблон:Math for Шаблон:Math applies, finally arriving at the complex integral definition of Шаблон:Math[1]
<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 Шаблон:Math and Шаблон:Mvar.
Limit for Шаблон:Math
Real values
Given the integral representation of a principal branch of Шаблон:Math, the following equation holds for all positive real Шаблон:Mvar, Шаблон:Mvar:[7] <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 Шаблон:Mvar. Assume first Шаблон:Math and Шаблон:Math. Then <math display="block">|\gamma(s, b) - \gamma(s, a)| \le \int_a^b |t^{s-1}| e^{-t}\, dt = \int_a^b t^{\Re s-1} e^{-t}\, dt \le \int_a^b t e^{-t}\, dt</math> where[8] <math display="block">|z^s| = |z|^{\Re s}\,e^{-\Im s\arg z}</math> has been used in the middle. Since the final integral becomes arbitrarily small if only Шаблон:Mvar is large enough, Шаблон:Math converges uniformly for Шаблон:Math on the strip Шаблон:Math towards a holomorphic function,[3] which must be Γ(s) because of the identity theorem. Taking the limit in the recurrence relation Шаблон:Math and noting, that lim Шаблон:Math for Шаблон:Math and all Шаблон:Mvar, shows, that Шаблон:Math 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 Шаблон:Mvar not a non-positive integer, Шаблон:Mvar real and Шаблон:Math principal.
Sectorwise convergence
Now let Шаблон:Mvar be from the sector Шаблон:Math with some fixed Шаблон:Mvar (Шаблон:Math), Шаблон:Math 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 Шаблон:Math is sufficiently large. The second difference allows for following estimation:
<math display="block">|\gamma(s, |u|) - \gamma(s, u)| \le \int_u^{|u|} |z^{s-1} e^{-z}|\, dz = \int_u^{|u|} |z|^{\Re s - 1}\,e^{-\Im s\,\arg z}\,e^{-\Re z} \, dz,</math>
where we made use of the integral representation of Шаблон:Math and the formula about Шаблон:Math above. If we integrate along the arc with radius Шаблон:Math around 0 connecting Шаблон:Mvar and Шаблон:Math, 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 Шаблон:Math is a constant independent of Шаблон:Mvar or Шаблон:Mvar. Again referring to the behavior of Шаблон:Math for large Шаблон:Mvar, we see that the last expression approaches 0 as Шаблон:Mvar increases towards Шаблон: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 Шаблон:Mvar is not a non-negative integer, Шаблон:Math is arbitrarily small, but fixed, and Шаблон:Math denotes the principal branch on this domain.
Overview
<math>\gamma(s, z)</math> is:
- entire in Шаблон:Mvar for fixed, positive integer Шаблон:Mvar;
- multi-valued holomorphic in Шаблон:Mvar for fixed Шаблон:Mvar not an integer, with a branch point at Шаблон:Math;
- on each branch meromorphic in Шаблон:Mvar for fixed Шаблон:Math, with simple poles at non-positive integers s.
Upper incomplete gamma function
As for the upper incomplete gamma function, a holomorphic extension, with respect to Шаблон:Mvar or Шаблон:Mvar, is given by[1] <math display="block">\Gamma(s,z) = \Gamma(s) - \gamma(s, z)</math> at points Шаблон:Math, 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 Шаблон:Mvar is a non-positive integer in the above equation, neither part of the difference is defined, and a limiting process, here developed for Шаблон:Math, fills in the missing values. Complex analysis guarantees holomorphicity, because <math>\Gamma(s,z)</math> proves to be bounded in a neighbourhood of that limit for a fixed Шаблон:Mvar.
To determine the limit, the power series of <math>\gamma^*</math> at Шаблон:Math is useful. When replacing <math>e^{-x}</math> by its power series in the integral definition of <math>\gamma</math>, one obtains (assume Шаблон:Mvar,Шаблон:Mvar 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 (-1)^k\,\frac{t^{s+k-1}}{k!} \, dt = \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>
or[4] <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 Шаблон:Mvar (and all complex Шаблон:Mvar 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{(-z)^k}{k!(s+k)},\quad \Re(s) > -1, \,s \ne 0.</math>
When Шаблон:Math:[9] <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 Шаблон:Math, also known as the exponential integral <math>E_1(z)</math>.[10]
By way of the recurrence relation, values of <math>\Gamma(-n, z)</math> for positive integers Шаблон:Mvar can be derived from this result,[11]
<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 + (-1)^n \Gamma(0, z)\right)</math>
so the upper incomplete gamma function proves to exist and be holomorphic, with respect both to Шаблон:Mvar and Шаблон:Mvar, for all Шаблон:Mvar and Шаблон:Math.
<math>\Gamma(s, z)</math> is:
- entire in Шаблон:Mvar for fixed, positive integral Шаблон:Mvar;
- multi-valued holomorphic in Шаблон:Mvar for fixed Шаблон:Mvar non zero and not a positive integer, with a branch point at Шаблон:Math;
- equal to <math>\Gamma(s)</math> for Шаблон:Mvar with positive real part and Шаблон:Math (the limit when <math>(s_i,z_i) \to (s, 0)</math>), but this is a continuous extension, not an analytic one (does not hold for real Шаблон:Math!);
- on each branch entire in Шаблон:Mvar for fixed Шаблон:Math.
Special values
- <math>\Gamma(s+1,1) = \frac{\lfloor es! \rfloor}{e} </math> if Шаблон:Mvar is a positive integer,
- <math>\Gamma(s,x) = (s-1)!\, e^{-x} \sum_{k=0}^{s-1} \frac{x^k}{k!}</math> if Шаблон:Mvar is a positive integer,[12]
- <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 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 Шаблон:Math, the error of Шаблон:Math is on the order of Шаблон:Math if Шаблон:Math and Шаблон:Math if Шаблон:Math),
- <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>.[13]
- <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.[13]
- <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>.[14]
Evaluation formulae
The lower gamma function can be evaluated using the power series expansion:[15] <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 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 Шаблон:Math is Kummer's confluent hypergeometric function.
Connection with Kummer's confluent hypergeometric function
When the real part of Шаблон:Mvar 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 Шаблон:Mvar, provided only that Шаблон:Mvar is not a negative integer.
The upper gamma function has the continued fraction[16] <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> andШаблон:Citation needed <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: <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 systems.
Even if unavailable directly, however, incomplete function values can be calculated using functions commonly included in spreadsheets (and computer algebra packages). In 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>
= EXP(GAMMALN(s))*GAMMA.DIST(x,s,1,TRUE). - The upper incomplete function: <math> \Gamma(s, x) </math>
= EXP(GAMMALN(s))*(1-GAMMA.DIST(x,s,1,TRUE)).
These follow from the definition of the gamma distribution's cumulative distribution function.
In Python, the Scipy library provides implementations of incomplete gamma functions under Шаблон:Code, however, it does not support negative values for the first argument. The function Шаблон:Code 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">P(s,x)=\frac{\gamma(s,x)}{\Gamma(s)},</math> <math display="block">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 random variables with shape parameter <math>s</math> and scale parameter 1.
When <math>s</math> is an integer, <math>Q(s, \lambda)</math> is the cumulative distribution function for Poisson random variables: If <math>X</math> is a <math>\mathrm{Poi}(\lambda)</math> random variable then
<math display="block"> \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.
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) =
\displaystyle\int_0^\infty e^{\left( -{x^s}/{\nu} \right)}
\, \mathfrak{N}_{{1}/{s}}\left(\nu\right) \, d\nu , \,\, (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 gammainc[17] and gammaincc[18] in scipy.
Derivatives
Using the integral representation above, the derivative of the upper incomplete gamma function <math> \Gamma (s,x) </math> with respect to Шаблон:Mvar 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[19] <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[\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 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{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 display="block">T(m,s,z) = - \frac{(-1)^{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{(-1)^n z^{s-1+n}}{n! (-s-n)^{m-1} }</math> with the understanding that Шаблон:Mvar 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.[20][21] 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 transforms and Mellin transforms. 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">\int x^{b-1} \gamma(s,x) dx = \frac{1}{b} \left( x^b \gamma(s,x) - \gamma(s+b,x) \right),</math> <math display="block">\int x^{b-1} \Gamma(s,x) dx = \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 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 Шаблон:Harv.
Notes
- ↑ 1,0 1,1 1,2 1,3 1,4 1,5 Шаблон:Cite web
- ↑ 2,0 2,1 2,2 Шаблон:Cite web
- ↑ 3,0 3,1 Шаблон:Cite web
- ↑ 4,0 4,1 Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ see last eq.
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Шаблон:Mathworld (equation 2)
- ↑ 13,0 13,1 Шаблон:Cite book
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ Abramowitz and Stegun p. 263, 6.5.31
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite web
- ↑ 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, [1]
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite arXiv, App B
References
- Шаблон:AS ref Шаблон:Cite web §6.5.
- Шаблон:Cite journal
- Шаблон:Cite journal
- G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal (See also www.netlib.org/toms/654).
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Springer
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Dlmf
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite journal
- Шаблон:Cite conference
- Шаблон:MathWorld
External links
- <math>P(a,x)</math> — Regularized Lower Incomplete Gamma Function Calculator
- <math>Q(a,x)</math> — Regularized Upper Incomplete Gamma Function Calculator
- <math>\gamma(a,x)</math> — Lower Incomplete Gamma Function Calculator
- <math>\Gamma(a,x)</math> — Upper Incomplete Gamma Function Calculator
- formulas and identities of the Incomplete Gamma Function functions.wolfram.com
_in_the_complex_plane_from_-2-2i_to_2%2B2i_with_colors_created_with_Mathematica_13.1_function_ComplexPlot3D.svg){kind=link}