Asymptotic expansions for the incomplete gamma function in the transition regions
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- by Gergő Nemes and Adri B. Olde Daalhuis HTML | PDF
- Math. Comp. 88 (2019), 1805-1827 Request permission
Abstract:
We construct asymptotic expansions for the normalised incomplete gamma function $Q(a,z)=\Gamma (a,z)/\Gamma (a)$ that are valid in the transition regions, including the case $z\approx a$, and have simple polynomial coefficients. For Bessel functions, these types of expansions are well known, but for the normalised incomplete gamma function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are again simple polynomials, and therefore its implementation is straightforward. As a byproduct, we give the first complete asymptotic expansion as $a\to -\infty$ of the unique negative zero of the regularised incomplete gamma function $\gamma ^*(a,x)$.References
- Stella Brassesco and Miguel A. Méndez, The asymptotic expansion for $n!$ and the Lagrange inversion formula, Ramanujan J. 24 (2011), no. 2, 219–234. MR 2765610, DOI 10.1007/s11139-010-9237-2
- R. B. Dingle, Asymptotic expansions: their derivation and interpretation, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1973. MR 0499926
- T. M. Dunster, Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities, Proc. Roy. Soc. London Ser. A 452 (1996), no. 1949, 1351–1367. MR 1406769, DOI 10.1098/rspa.1996.0069
- T. M. Dunster, R. B. Paris, and S. Cang, On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function, Methods Appl. Anal. 5 (1998), no. 3, 223–247. MR 1659076, DOI 10.4310/MAA.1998.v5.n3.a1
- Walter Gautschi, Exponential integral $\int _{1}{}^{\infty }e ^{-xt}t^{-n}dt$ for large values of $n$, J. Res. Nat. Bur. Standards 62 (1959), 123–125. MR 0104347, DOI 10.6028/jres.062.022
- Michael B. Giles, Algorithm 955: approximation of the inverse Poisson cumulative distribution function, ACM Trans. Math. Software 42 (2016), no. 1, Art. 7, 22. MR 3472423, DOI 10.1145/2699466
- Chelo Ferreira, José L. López, and Ester Pérez Sinusía, Incomplete gamma functions for large values of their variables, Adv. in Appl. Math. 34 (2005), no. 3, 467–485. MR 2123546, DOI 10.1016/j.aam.2004.08.001
- Henry E. Fettis, An asymptotic expansion for the upper percentage points of the $\chi ^{2}$-distribution, Math. Comp. 33 (1979), no. 147, 1059–1064. MR 528059, DOI 10.1090/S0025-5718-1979-0528059-9
- Norman L. Johnson, Samuel Kotz, and N. Balakrishnan, Continuous univariate distributions. Vol. 1, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Inc., New York, 1994. A Wiley-Interscience Publication. MR 1299979
- K. S. Kölbig, On the zeros of the incomplete gamma function, Math. Comp. 26 (1972), 751–755. MR 326994, DOI 10.1090/S0025-5718-1972-0326994-1
- Mark A. Lukas, Performance criteria and discrimination of extreme undersmoothing in nonparametric regression, J. Statist. Plann. Inference 153 (2014), 56–74. MR 3229022, DOI 10.1016/j.jspi.2014.05.006
- K. Mahler, Über die Nullstellen der unvollständigen Gammafunktionen, Rend. del Circ. Matem. Palermo 54 (1930), 1–41.
- Claude Mitschi and David Sauzin, Divergent series, summability and resurgence. I, Lecture Notes in Mathematics, vol. 2153, Springer, [Cham], 2016. Monodromy and resurgence; With a foreword by Jean-Pierre Ramis and a preface by Éric Delabaere, Michèle Loday-Richaud, Claude Mitschi and David Sauzin. MR 3526111, DOI 10.1007/978-3-319-28736-2
- A. Navarra, C. M. Pinotti, V. Ravelomanana, F. Betti Sorbelli, and R. Ciotti, Cooperative training for high density sensor and actor networks, J. Sel. Areas Commun. 28 (2010), no. 5, 753–763.
- Gergő Nemes, An explicit formula for the coefficients in Laplace’s method, Constr. Approx. 38 (2013), no. 3, 471–487. MR 3122279, DOI 10.1007/s00365-013-9202-6
- Gergő Nemes, The resurgence properties of the incomplete gamma function II, Stud. Appl. Math. 135 (2015), no. 1, 86–116. MR 3366821, DOI 10.1111/sapm.12077
- Gergő Nemes, The resurgence properties of the incomplete gamma function, I, Anal. Appl. (Singap.) 14 (2016), no. 5, 631–677. MR 3530271, DOI 10.1142/S0219530515500128
- NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.19 of 2018-06-22. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, and B. V. Saunders, eds.
- A. B. Olde Daalhuis, On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function, Methods Appl. Anal. 5 (1998), no. 4, 425–438. MR 1669847, DOI 10.4310/MAA.1998.v5.n4.a7
- V. I. Pagurova, An asymptotic formula for the incomplete gamma function, U.S.S.R. Comput. Math. and Math. Phys. 5 (1965), no. 1, pp. 162–166.
- G. Paillard and V. Ravelomanana, Limit theorems for degree of coverage and lifetime in large sensor networks, IEEE INFOCOM 2008 - The 27th Conference on Computer Communications, Phoenix, Arizona, 2008, pp. 106–110.
- P. Palffy-Muhoray, E. G. Virga, and X. Zheng, Onsager’s missing steps retraced, J. Phys.: Condens. Matter 29 (2017), no. 47, Article 475102, 13 pp.
- R. B. Paris, Error bounds for the uniform asymptotic expansion of the incomplete gamma function, J. Comput. Appl. Math. 147 (2002), no. 1, 215–231. MR 1931777, DOI 10.1016/S0377-0427(02)00434-X
- R. B. Paris, A uniform asymptotic expansion for the incomplete gamma function, J. Comput. Appl. Math. 148 (2002), no. 2, 323–339. MR 1936142, DOI 10.1016/S0377-0427(02)00553-8
- R. B. Paris, A uniform asymptotic expansion for the incomplete gamma functions revisited, preprint, arXiv:1611.00548
- V. Ravelomanana, Extremal properties of three-dimensional sensor networks with applications, IEEE Trans. Mobile Comput. 3 (2004), no. 3, 246–257.
- N. M. Temme, The asymptotic expansion of the incomplete gamma functions, SIAM J. Math. Anal. 10 (1979), no. 4, 757–766. MR 533947, DOI 10.1137/0510071
- N. M. Temme, The uniform asymptotic expansion of a class of integrals related to cumulative distribution functions, SIAM J. Math. Anal. 13 (1982), no. 2, 239–253. MR 647123, DOI 10.1137/0513017
- N. M. Temme, Asymptotic inversion of incomplete gamma functions, Math. Comp. 58 (1992), no. 198, 755–764. MR 1122079, DOI 10.1090/S0025-5718-1992-1122079-8
- N. M. Temme, Computational aspects of incomplete gamma functions with large complex parameters, Approximation and computation (West Lafayette, IN, 1993) Internat. Ser. Numer. Math., vol. 119, Birkhäuser Boston, Boston, MA, 1994, pp. 551–562. MR 1333643
- N. M. Temme, Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters, Methods Appl. Anal. 3 (1996), no. 3, 335–344. MR 1421474, DOI 10.4310/MAA.1996.v3.n3.a3
- Ian Thompson, A note on the real zeros of the incomplete gamma function, Integral Transforms Spec. Funct. 23 (2012), no. 6, 445–453. MR 2929187, DOI 10.1080/10652469.2011.597391
- Lidija Trailović and Lucy Y. Pao, Computing budget allocation for efficient ranking and selection of variances with application to target tracking algorithms, IEEE Trans. Automat. Control 49 (2004), no. 1, 58–67. MR 2028542, DOI 10.1109/TAC.2003.821428
- Lloyd N. Trefethen and J. A. C. Weideman, The exponentially convergent trapezoidal rule, SIAM Rev. 56 (2014), no. 3, 385–458. MR 3245858, DOI 10.1137/130932132
- F. G. Tricomi, Asymptotische Eigenschaften der unvollständigen Gammafunktion, Math. Z. 53 (1950), 136–148 (German). MR 45253, DOI 10.1007/BF01162409
Additional Information
- Gergő Nemes
- Affiliation: School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
- Email: gergo.nemes@ed.ac.uk
- Adri B. Olde Daalhuis
- Affiliation: School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
- MR Author ID: 293428
- Email: a.oldedaalhuis@ed.ac.uk
- Received by editor(s): March 21, 2018
- Received by editor(s) in revised form: May 11, 2018
- Published electronically: November 8, 2018
- Additional Notes: The authors’ research was supported by a research grant (GRANT11863412/70NANB15H221) from the National Institute of Standards and Technology.
- © Copyright 2018 American Mathematical Society
- Journal: Math. Comp. 88 (2019), 1805-1827
- MSC (2010): Primary 33B20, 41A60
- DOI: https://doi.org/10.1090/mcom/3391
- MathSciNet review: 3925486