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 edit
The upper incomplete gamma function is defined as:
Properties edit
By integration by parts we find the recurrence relations
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: [2]
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;[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 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 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. 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 (α − δ, 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 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, 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 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 of ,[1] for k a suitable integer.
Behavior near branch point edit
The decomposition above further shows, that γ behaves near z = 0 asymptotically like:
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, stating that equations between holomorphic functions valid on a real interval, hold everywhere. In particular, the recurrence relation [2] and ∂γ(s, z)/∂z = zs−1 e−z [2] 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 e−z. Consequently, for any complex u, v ≠ 0,
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:[7]
s complex edit
This result extends to complex s. Assume first 1 ≤ Re(s) ≤ 2 and 1 < a < b. Then
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
As shown above, the first difference can be made arbitrarily small, if |u| is sufficiently large. The second difference allows for following estimation:
Overview edit
is:
- entire in z for fixed, positive integer 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[1]
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 proves to be bounded in a neighbourhood of that limit for a fixed z.
To determine the limit, the power series of at z = 0 is useful. When replacing by its power series in the integral definition of , one obtains (assume x,s positive reals for now):
With its restriction to real values lifted, the series allows the expansion:
When s → 0:[9]
By way of the recurrence relation, values of for positive integers n can be derived from this result,[11]
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;
- equal to for s with positive real part and z = 0 (the limit when ), 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
Here, is the exponential integral, is the generalized exponential integral, is the error function, and is the complementary error function, .
Asymptotic behavior edit
- as ,
- as and (for real s, the error of Γ(s, x) ~ −xs / s is on the order of O(xmin{s + 1, 0}) if s ≠ −1 and O(ln(x)) if s = −1),
- as an asymptotic series where and .[13]
- as an asymptotic series where and , where , where is the Euler-Mascheroni constant.[13]
- as ,
- as ,
- as an asymptotic series where and .[14]
Evaluation formulae edit
The lower gamma function can be evaluated using the power series expansion:[15]
An alternative expansion is
Connection with Kummer's confluent hypergeometric function edit
When the real part of z is positive,
Again with confluent hypergeometric functions and employing Kummer's identity,
For the actual computation of numerical values, Gauss's continued fraction provides a useful expansion:
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[16]
Multiplication theorem edit
The following multiplication theorem holds true:
Software implementation edit
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:
= EXP(GAMMALN(s))*GAMMA.DIST(x,s,1,TRUE)
. - The upper incomplete function:
= 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 scipy.special
, however, it does not support negative values for the first argument. The function gammainc
from the mpmath library supports all complex arguments.
Regularized gamma functions and Poisson random variables edit
Two related functions are the regularized gamma functions:
When is an integer, is the cumulative distribution function for Poisson random variables: If is a random variable then
This formula can be derived by repeated integration by parts.
In the context of the stable count distribution, the parameter can be regarded as inverse of Lévy's stability parameter :
and are implemented as gammainc
[17] and gammaincc
[18] in scipy.
Derivatives edit
Using the integral representation above, the derivative of the upper incomplete gamma function with respect to x is
Indefinite and definite integrals edit
The following indefinite integrals are readily obtained using integration by parts (with the constant of integration omitted in both cases):
Notes edit
- ^ a b c d e f "DLMF: §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions". dlmf.nist.gov.
- ^ a b c "DLMF: §8.8 Recurrence Relations and Derivatives ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions". dlmf.nist.gov.
- ^ a b Donald E. Marshall (Autumn 2009). "Complex Analysis" (PDF). Math 534 (student handout). University of Washington. Theorem 3.9 on p.56. Archived from the original (PDF) on 16 May 2011. Retrieved 23 April 2011.
- ^ a b "DLMF: §8.7 Series Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions". dlmf.nist.gov.
- ^ Paul Garrett. "Hartogs' Theorem: separate analyticity implies joint" (PDF). cse.umn.edu. Retrieved 21 December 2023.
- ^ C. Teleman. "Riemann Surfaces" (PDF). berkeley.edu. Retrieved 21 December 2023.
- ^ "DLMF: §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function". dlmf.nist.gov.
- ^ "DLMF: §4.4 Special Values and Limits ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions". dlmf.nist.gov.
- ^ see last eq.
- ^ "DLMF: §8.4 Special Values ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions". dlmf.nist.gov.
- ^ "DLMF: 8.4 Special Values".
- ^ Weisstein, Eric W. "Incomplete Gamma Function". MathWorld. (equation 2)
- ^ a b Bender & Orszag (1978). Advanced Mathematical Methods for Scientists and Engineers. Springer.
- ^ "DLMF: §8.11 Asymptotic Approximations and Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions". dlmf.nist.gov.
- ^ "DLMF: §8.11 Asymptotic Approximations and Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions". dlmf.nist.gov.
- ^ Abramowitz and Stegun p. 263, 6.5.31
- ^ "scipy.special.gammainc — SciPy v1.11.4 Manual". docs.scipy.org.
- ^ "scipy.special.gammaincc — SciPy v1.11.4 Manual". docs.scipy.org.
- ^ 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]
- ^ Milgram, M. S. (1985). "The generalized integro-exponential function". Math. Comp. 44 (170): 443–458. doi:10.1090/S0025-5718-1985-0777276-4. MR 0777276.
- ^ 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
- Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 6.5". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. "Incomplete Gamma function". §6.5.
- Allasia, Giampietro; Besenghi, Renata (1986). "Numerical calculation of incomplete gamma functions by the trapezoidal rule". Numer. Math. 50 (4): 419–428. doi:10.1007/BF01396662. S2CID 121964300.
- Amore, Paolo (2005). "Asymptotic and exact series representations for the incomplete Gamma function". Europhys. Lett. 71 (1): 1–7. arXiv:math-ph/0501019. Bibcode:2005EL.....71....1A. doi:10.1209/epl/i2005-10066-6. MR 2170316. S2CID 1921569.
- G. Arfken and H. Weber. Mathematical Methods for Physicists. Harcourt/Academic Press, 2000. (See Chapter 10.)
- DiDonato, Armido R.; Morris, Jr., Alfred H. (December 1986). "Computation of the incomplete gamma function ratios and their inverse". ACM Transactions on Mathematical Software. 12 (4): 377–393. doi:10.1145/22721.23109. S2CID 14351930.
- Barakat, Richard (1961). "Evaluation of the Incomplete Gamma Function of Imaginary Argument by Chebyshev Polynomials". Math. Comp. 15 (73): 7–11. doi:10.1090/s0025-5718-1961-0128058-1. MR 0128058.
- Carsky, Petr; Polasek, Martin (1998). "Incomplete Gamma F_m(x) functions for real and complex arguments". J. Comput. Phys. 143 (1): 259–265. Bibcode:1998JCoPh.143..259C. doi:10.1006/jcph.1998.5975. MR 1624704.
- Chaudhry, M. Aslam; Zubair, S. M. (1995). "On the decomposition of generalized incomplete Gamma functions with applications to Fourier transforms". J. Comput. Appl. Math. 59 (101): 253–284. doi:10.1016/0377-0427(94)00026-w. MR 1346414.
- DiDonato, Armido R.; Morris, Jr., Alfred H. (September 1987). "ALGORITHM 654: FORTRAN subroutines for computing the incomplete gamma function ratios and their inverse". ACM Transactions on Mathematical Software. 13 (3): 318–319. doi:10.1145/29380.214348. S2CID 19902932. (See also www.netlib.org/toms/654).
- 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. doi:10.1145/198429.198432. S2CID 16737306.
- Gautschi, Walter (1998). "The incomplete gamma function since Tricomi". Atti Convegni Lincei. 147: 203–237. MR 1737497.
- Gautschi, Walter (1999). "A Note on the recursive calculation of Incomplete Gamma Functions". ACM Trans. Math. Softw. 25 (1): 101–107. doi:10.1145/305658.305717. MR 1697463. S2CID 36469885.
- Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "8.35.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. pp. 908–911. ISBN 978-0-12-384933-5. LCCN 2014010276.
- Jones, William B.; Thron, W. J. (1985). "On the computation of incomplete gamma functions in the complex domain". J. Comput. Appl. Math. 12–13: 401–417. doi:10.1016/0377-0427(85)90034-2. MR 0793971.
- "Incomplete gamma-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Mathar, Richard J. (2004). "Numerical representation of the incomplete gamma function of complex-valued argument". Numerical Algorithms. 36 (3): 247–264. arXiv:math/0306184. Bibcode:2004NuAlg..36..247M. doi:10.1023/B:NUMA.0000040063.91709.58. MR 2091195. S2CID 30860614.
- Miller, Allen R.; Moskowitz, Ira S. (1998). "On certain Generalized incomplete Gamma functions". J. Comput. Appl. Math. 91 (2): 179–190. doi:10.1016/s0377-0427(98)00031-4.
- Paris, R. B. (2010), "Incomplete gamma function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
- Paris, R. B. (2002). "A uniform asymptotic expansion for the incomplete gamma function". J. Comput. Appl. Math. 148 (2): 323–339. Bibcode:2002JCoAM.148..323P. doi:10.1016/S0377-0427(02)00553-8. MR 1936142.
- Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 6.2. Incomplete Gamma Function and Error Function". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.
- Takenaga, Roy (1966). "On the Evaluation of the Incomplete Gamma Function". Math. Comp. 20 (96): 606–610. doi:10.1090/S0025-5718-1966-0203911-3. MR 0203911.
- Temme, Nico (1975). "Uniform Asymptotic Expansions of the Incomplete Gamma Functions and the Incomplete Beta Function". Math. Comp. 29 (132): 1109–1114. doi:10.1090/S0025-5718-1975-0387674-2. MR 0387674.
- Terras, Riho (1979). "The determination of incomplete Gamma Functions through analytic integration". J. Comput. Phys. 31 (1): 146–151. Bibcode:1979JCoPh..31..146T. doi:10.1016/0021-9991(79)90066-4. MR 0531128.
- Tricomi, Francesco G. (1950). "Sulla funzione gamma incompleta". Ann. Mat. Pura Appl. 31: 263–279. doi:10.1007/BF02428264. MR 0047834. S2CID 120404791.
- Tricomi, F. G. (1950). "Asymptotische Eigenschaften der unvollst. Gammafunktion". Math. Z. 53 (2): 136–148. doi:10.1007/bf01162409. MR 0045253. S2CID 121234109.
- van Deun, Joris; Cools, Ronald (2006). "A stable recurrence for the incomplete gamma function with imaginary second argument". Numer. Math. 104 (4): 445–456. doi:10.1007/s00211-006-0026-1. MR 2249673. S2CID 43780150.
- Winitzki, Serge (2003). "Computing the Incomplete Gamma Function to Arbitrary Precision". In Vipin Kumar; Marina L. Gavrilova; Chih Jeng Kenneth Tan; Pierre L'Ecuyer (eds.). Computational Science and Its Applications — ICSSA 2003. International Conference on Computational Science and Its Applications, Montreal, Canada, May 18–21, 2003, Proceedings, Part I. Lecture Notes in Computer Science. Vol. 2667. pp. 790–798. doi:10.1007/3-540-44839-x_83. ISBN 978-3-540-40155-1. MR 2110953.
- Weisstein, Eric W. "Incomplete Gamma Function". MathWorld.