Incomplete gamma function

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In mathematics, the upper incomplete gamma function and lower incomplete gamma function 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, another type of special function, but with different or "incomplete" integral limits. More specifically, the upper incomplete function has in its integral definition an upper integral limit that is not necessarily infinite, as opposed to the gamma function, but some positive value. Similarly, the lower incomplete gamma function has a non-zero lower integral limit, which may not be zero. These respective definitions contrast with the gamma function whose integral definition has the limits of zero and infinity.


Definition[edit]

The upper incomplete function is defined as:

 \Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t ,\,\!

whereas the lower incomplete function is defined as:

  \gamma(s,x) = \int_0^x t^{s-1}\,e^{-t}\,{\rm d}t .\,\!

Properties[edit]

In both cases s is a complex parameter, such that the real part of s is positive.

By integration by parts we find the recurrence relations

\Gamma(s,x)= (s-1)\Gamma(s-1,x) + x^{s-1} e^{-x}

and conversely

 \gamma(s,x) =(s-1)\gamma(s-1,x) - x^{s-1} e^{-x}

Since the ordinary gamma function is defined as

 \Gamma(s) = \int_0^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t

we have

 \Gamma(s) = \Gamma(s,0)

and

 \gamma(s,x) + \Gamma(s,x) = \Gamma(s).

Continuation to complex values[edit]

The lower incomplete gamma and the upper incomplete gamma function, as defined above for real positive s and x, can be developed into holomorphic functions, with respect both to x and s, defined for almost all combinations of complex x and s.[1] Complex analysis shows how properties of the real incomplete gamma functions extend to their holomorphic counterparts.

Lower incomplete Gamma function[edit]

Holomorphic extension[edit]

Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: [4]

\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)}

Given the rapid growth in absolute value of Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all complex s and x. By a theorem of Weierstraß,[2] the limiting function, sometimes denoted as \gamma^*,

\gamma^*(s, z) := e^{-z}\sum_{k=0}^\infty\frac{z^k}{\Gamma(s+k+1)} [5]

is entire with respect to both z (for fixed s) and s (for fixed z) [6], and, thus, holomorphic on ℂ×ℂ by Hartog's theorem[7]. Hence, the following decomposition

\gamma(s,z) = z^s \, \Gamma(s) \, \gamma^*(s,z) [8],

extends the real lower incomplete gamma function as a holomorphic function, both jointly and separately in z and s. It follows from the properties of zs and the Γ-function, that the first two factors capture the singularities of γ (at z = 0 or s a non-positive integer), whereas the last factor contributes to its zeros.

Multi-valuedness[edit]

The complex logarithm log z = log |z| + i arg z is determined up to a multiple of 2πi only, which renders it multi-valued. Functions involving the complex logarithm typically inherit this property. Among these are the complex power, and, since zs appears in its decomposition, the 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 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 [9];
  • 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[edit]

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 αδ < arg z < α + δ with some α and 0 < δπ. 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. Note: In many applications and texts, α is silently taken to be 0, which centers the sector around the positive real axis.

Branches[edit]

In particular, a single-valued and holomorphic logarithm exists on any such sector D having its imaginary part bound to the range (αδ, α + δ). Based on such a restricted logarithm, zs and the incomplete gamma functions in turn collapse to single-valued, holomorphic functions on D (or ×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, because they equal their real analogons on the positive real axis. Note: In many applications and texts, formulas hold only for principal branches.

The exponential function es[edit]

The expression es shall always denote the exponential function, which is the restriction of a principal branch of zs to z = e.

Relation between branches[edit]

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 e^{s*2k\pi i}[10], k a suitable integer.

Behavior near branch point[edit]

The decomposition above further shows, that γ behaves near z = 0 asymptotically like:

\gamma(s, z) \asymp z^s \, \Gamma(s) \, \gamma^*(s, 0) = z^s \, \Gamma(s)/\Gamma(s+1) = z^s/s

For positive real x, y and s, xy/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 uv 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 of γ(b) is naturally fulfilled, so there γ(s, 0) = 0 for s with positive real part is a continuous limit. Also note that such a continuation is by no means an analytic one.

Algebraic relations[edit]

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 [11], stating that equations between holomorphic functions valid on a real interval, hold everywhere. In particular, the recurrence relation [12] and ∂γ(s,z)/∂z = zs−1 ez [13] are preserved on corresponding branches.

Integral representation[edit]

The last relation tells us, that, for fixed s, γ is a primitive or antiderivative of the holomorphic function zs−1 ez. Consequently [14], for any complex u, v ≠ 0,

\int_u^v t^{s-1}\,e^{-t}\,{\rm d}t = \gamma(s,v) - \gamma(s,u)

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 s is positive, then the limit γ(s, u) → 0 for u → 0 applies, finally arriving at the complex integral definition of γ

\gamma(s, z) = \int_0^z t^{s-1}\,e^{-t}\,{\rm d}t, \, \Re(s) > 0. [15]

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.

Limit for z → +∞[edit]
Real values[edit]

Given the integral representation of a principal branch of γ, the following equation holds for all positive real s, x:[16]

\Gamma(s) = \int_0^\infty t^{s-1}\,e^{-t}\,{\rm d}t = \lim_{x \rightarrow \infty} \gamma(s, x)
s complex[edit]

This result extends to complex s. Assume first 1 ≤ Re(s) ≤ 2 and 1 < a < b. Then

|\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

where

|z^s| = |z|^{\Re s}\,e^{-\Im s\arg z}[17]

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 1 ≤ Re(s) ≤ 2 towards a holomorphic function,[3] which must be Γ(s) because of the identity theorem [18]. Taking the limit in the recurrence relation γ(s,x) = (s − 1)γ(s − 1,x) − xs −1 ex and noting, that lim xn ex = 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

\Gamma(s) = \lim_{x \rightarrow \infty} \gamma(s, x)

for all complex s not a non-positive integer, x real and γ principal.

Sectorwise convergence[edit]

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

\Gamma(s) - \gamma(s, u) = \Gamma(s) - \gamma(s, |u|) + \gamma(s, |u|) - \gamma(s, u).

As shown above, the first difference can be made arbitrarily small, if |u| is sufficiently large. The second difference allows for following estimation:

|\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

where we made use of the integral representation of γ and the formula about |zs| above. If we integrate along the arc with radius R = |u| around 0 connecting u and |u|, then the last integral is

\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}

where M = δ(cos δ)−Re s eIm is a constant independent of u or R. Again referring to the behavior of xn ex for large x, we see that the last expression approaches 0 as R increases towards ∞. In total we now have:

\Gamma(s) = \lim_{|z| \rightarrow \infty} \gamma(s, z), \quad |\arg z| < \pi/2 - \epsilon

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.

Overview[edit]

\gamma(s, z) is:

  • entire in z for fixed, positive integral s;
  • multi-valued holomorphic in z for fixed s not an integer, with a branch point at z = 0;
  • on each branch meromorphic in s for fixed z ≠ 0, with simple poles at non-positive integers s.

Upper incomplete Gamma function[edit]

As for the upper incomplete gamma function, a holomorphic extension, with respect to z or s, is given by

\Gamma(s,z) = \Gamma(s) - \gamma(s, z) [19]

at points (s, z), where the right hand side exists. Since \gamma is multi-valued, the same holds for \Gamma, but a restriction to principal values only yields the single-valued principal branch of \Gamma.

When s is a non-positive integer in the above equation, neither part of the difference is defined, and a limiting process, here developed for s → 0, fills in the missing values. Complex analysis guarantees holomorphicity, because \Gamma(s,z) proves to be bounded in a neighbourhood of that limit for a fixed z[20].

To determine the limit, the power series of \gamma^* at z = 0 turns out useful. When replacing e^{-x} by its power series in the integral definition of \gamma, one obtains (assume x,s positive reals for now):

\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)}

or

\gamma^*(s,x) = \sum_{k=0}^\infty \frac{(-x)^k}{k!\,\Gamma(s)(s+k)}. [21]

which, as a series representation of the entire \gamma^* function, converges for all complex x (and all complex s not a non-positive integer).

With its restriction to real values lifted, the series allows the expansion:

\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

When s → 0:

\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,[4]

(\gamma is the Euler–Mascheroni constant here), hence,

\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!)}

is the limiting function to the upper incomplete gamma function as s → 0, also known as E_1(z).[5]

By way of the recurrence relation, values of \Gamma(-n, z) 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.

\Gamma(s, z) is:

  • entire in z for fixed, positive integral s;
  • multi-valued holomorphic in z for fixed s non zero and not a positive integer, with a branch point at z = 0;
  • = \Gamma(s) for s with positive real part and z = 0 (the limit when (s_i,z_i)  \rightarrow (s, 0)), but this is a continuous extension, not an analytic one (does not hold for real s<0!);
  • on each branch entire in s for fixed z ≠ 0.

Special values[edit]

  •  \Gamma(s,0) = \Gamma(s), \Re(s) > 0
  • \Gamma(1,x) = e^{-x},
  • \gamma(1,x) = 1 - e^{-x},
  • \Gamma(0,x) = -{\rm Ei}(-x) for x>0,
  • \Gamma(s,x) = x^s \, {\rm E}_{1-s}(x),
  • \Gamma\left(\tfrac{1}{2}, x\right) = \sqrt\pi\,{\rm erfc}\left(\sqrt x\right),
  • \gamma\left(\tfrac{1}{2}, x\right) = \sqrt\pi\,{\rm erf}\left(\sqrt x\right).

Here, \mathrm{Ei} is the exponential integral, \mathrm{E_n} is the generalized exponential integral, \mathrm{erf} is the error function, and \mathrm{erfc} is the complementary error function, \operatorname{erfc}(x) = 1-\operatorname{erf}(x).

Asymptotic behavior[edit]

  •  \frac{\gamma(s,x)}{x^s} \rightarrow \frac 1 s as x \rightarrow 0,
  •  \frac{\Gamma(s,x)}{x^s} \rightarrow -\frac 1 s as x \rightarrow 0 and \Re (s) < 0\,
  •  \gamma(s,x) \rightarrow \Gamma(s) as x \rightarrow \infty,
  •  \frac{\Gamma(s,x)}{x^{s-1} e^{-x}} \rightarrow 1 as x \rightarrow \infty,

Evaluation formulae[edit]

The lower gamma function has the straight forward expansion

\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),

where M is Kummer's confluent hypergeometric function.

Connection with Kummer's confluent hypergeometric function[edit]

When the real part of z is positive,


\gamma(s,z) = \frac{}{} s^{-1} z^s e^{-z} M(1,s+1,z)

where


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

has an infinite radius of convergence.

Again with confluent hypergeometric functions and employing Kummer's identity,


\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}

For the actual computation of numerical values, Gauss's continued fraction provides a useful expansion:


\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}}}}}}}.

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


\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}}}}}}
[8]

and


\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}}}}}
[citation needed]

Multiplication theorem[edit]

The following multiplication theorem holds true:


\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}

Regularized Gamma functions and Poisson random variables[edit]

Two related functions are the regularized Gamma functions:

P(s,x)=\frac{\gamma(s,x)}{\Gamma(s)},
Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}=1-P(s,x).

P(s,x) is the cumulative distribution function for Gamma random variables with shape parameter s and scale parameter 1.

When s>0 is an integer, Q(s,\lambda) is the cumulative distribution function for Poisson random variables: If X is a {\rm Poi}(\lambda) random variable then

 Pr(X<s) = \sum_{i<s} e^{-\lambda} \frac{\lambda^i}{i!} = \frac{\Gamma(s,\lambda)}{\Gamma(s)} = Q(s,\lambda).

This formula can be derived by repeated integration by parts.

Derivatives[edit]

The derivative of the upper incomplete gamma function  \Gamma (s,x) with respect to x is well known. It is simply given by the integrand of its integral definition:

 
\frac{\partial \Gamma (s,x) }{\partial x} = - \frac{x^{s-1}}{e^x}

The derivative with respect to its first argument s is given by[9]


\frac{\partial \Gamma (s,x) }{\partial s} = \ln x \Gamma (s,x) + x\,T(3,s,x)

and the second derivative by


\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) ]

where the function T(m,s,x) is a special case of the Meijer G-function


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).

This particular special case has internal closure properties of its own because it can be used to express all successive derivatives. In general,


\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)

where  P_j^n is the permutation defined by the Pochhammer symbol:


P_j^n = \left( \begin{array}{l} n \\ j \end{array} \right) j! = \frac{n!}{(n-j)!}.

All such derivatives can be generated in succession from:


\frac{\partial T (m,s,x) }{\partial s} = \ln x ~ T(m,s,x) + (m-1) T(m+1,s,x)

and


\frac{\partial T (m,s,x) }{\partial x} = -\frac{1}{x} [T(m-1,s,x) + T(m,s,x)]

This function T(m,s,x) can be computed from its series representation valid for  |z| < 1 ,


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} }

with the understanding that s is not a negative integer or zero. In such a case, one must use a limit. Results for  |z| \ge 1 can be obtained by analytic continuation. Some special cases of this function can be simplified. For example, T(2,s,x)=\Gamma(s,x)/x, x\,T(3,1,x) = {\rm E}_1(x), where {\rm E}_1(x) is the Exponential integral. These derivatives and the function T(m,s,x) provide exact solutions to a number of integrals by repeated differentiation of the integral definition of the upper incomplete gamma function.[10][11] For example,

 
\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)

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[edit]

The following indefinite integrals are readily obtained using integration by parts:


\int x^{b-1} \gamma(s,x) \mathrm d x = \frac{1}{b} \left( x^b \gamma(s,x) + \Gamma(s+b,x) \right).

\int x^{b-1} \Gamma(s,x) \mathrm d x = \frac{1}{b} \left( x^b \Gamma(s,x) - \Gamma(s+b,x) \right),

The lower and the upper incomplete Gamma function are connected via the Fourier transform:


\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}.

This follows, for example, by suitable specialization of (Gradshteyn & Ryzhik 1980, § 7.642).

Notes[edit]

  1. ^ DLMF, Incomplete Gamma functions, analytic continuation
  2. ^ [1] Theorem 3.9 on p.56
  3. ^ [2] Theorem 3.9 on p.56
  4. ^ see last eq.
  5. ^ http://dlmf.nist.gov/8.4.E4
  6. ^ Weisstein, Eric W., "Incomplete Gamma Function", MathWorld. (equation 2)
  7. ^ DLMF, Incomplete Gamma functions, 8.11(i)
  8. ^ Abramowitz and Stegun p. 263, 6.5.31
  9. ^ 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, [3]
  10. ^ Milgram, M. S. Milgram (1985). "The generalized integro-exponential function". Math. Comp. 44 (170): 443–458. doi:10.1090/S0025-5718-1985-0777276-4. MR 0777276. 
  11. ^ Mathar (2009). "Numerical Evaluation of the Oscillatory Integral over exp(i*pi*x)*x^(1/x) between 1 and infinity". arXiv:0912.3844 [math.CA]., App B

References[edit]

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  • Allasia, Giampietro; Besenghi, Renata (1986). "Numerical calculation of incomplete gamma functions by the trapezoidal rule". Numer. Math. 50 (4): 419–428. doi:10.1007/BF0139666. 
  • Amore, Paolo (2005). "Asymptotic and exact series representations for the incomplete Gamma function". Europhys. Lett. 71 (1): 1–7. MR 2170316. 
  • G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)
  • DiDonato, Armido R.; Morris, Jr., Alfred H. (Dec 1986). "Computation of the incomplete gamma function ratios and their inverse". ACM Transactions on Mathematical Software (TOMS) 12 (4): 377–393. doi:10.1145/22721.23109. 
  • Barakat, Richard (1961). "Evaluation of the Incomplete Gamma Function of Imaginary Argument by Chebyshev Polynomials". Math. Comp. 15 (73): 7–11. MR 0128058. 
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  • Früchtl, H.; Otto, P. (1994). "A new algorithm for the evaluation of the incomplete Gamma Function on vector computers". ACM Trans. Math. Softw. 20 (4): 436–446. 
  • Gautschi, Walter (1998). "The incomplete gamma function since Tricomi". Atti Convegni Lincei 147: 203–237. MR 1737497. 
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