Resurgence in the Transition Region: The Incomplete Gamma Function

Gergő Nemes

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

We study the resurgence properties of the coefficients $C_n(\tau)$ appearing in the asymptotic expansion of the incomplete gamma function within the transition region. Our findings reveal that the asymptotic behaviour of $C_n(\tau)$ as $n\to +\infty$ depends on the parity of $n$. Both $C_{2n-1}(\tau)$ and $C_{2n}(\tau)$ exhibit behaviours characterised by a leading term accompanied by an inverse factorial series, where the coefficients are once again $C_{2k-1}(\tau)$ and $C_{2k}(\tau)$, respectively. Our derivation employs elementary tools and relies on the known resurgence properties of the asymptotic expansion of the gamma function and the uniform asymptotic expansion of the incomplete gamma function. To the best of our knowledge, prior to this paper, there has been no investigation in the existing literature regarding the resurgence properties of asymptotic expansions in transition regions.

Détail

Keywords: asymptotic expansions, incomplete gamma function, resurgence, transition regions.

Bibliographie

[1] Bennett T., Howls C.J., Nemes G., Olde Daalhuis A.B., “Globally exact asymptotics for integrals with arbitrary order saddles”, SIAM J. Math. Anal., 50 (2018), 2144–2177, arXiv: 1710.10073 | DOI | MR | Zbl

[2] Berry M.V., Howls C.J., “Hyperasymptotics for integrals with saddles”, Proc. Roy. Soc. London Ser. A, 434 (1991), 657–675 | DOI | MR | Zbl

[3] Boyd W.G.C., “Gamma function asymptotics by an extension of the method of steepest descents”, Proc. Roy. Soc. London Ser. A, 447 (1994), 609–630 | DOI | MR | Zbl

[4] Dingle R.B., Asymptotic expansions: their derivation and interpretation, Academic Press, London, 1973 | MR | Zbl

[5] Dunster T.M., Paris R.B., Cang S., “On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function”, Methods Appl. Anal., 5 (1998), 223–247 | DOI | MR | Zbl

[6] Écalle J., Les fonctions résurgentes, v. I, Publ. Math. Orsay, 5, Resurgent functions, Université de Paris-Sud, Orsay, 1981 | MR

[7] Écalle J., Les fonctions résurgentes, v. II, Publ. Math. Orsay, 6, Les fonctions résurgentes appliquées à l'itération, Université de Paris-Sud, Orsay, 1981 | MR

[8] Écalle J., Les fonctions résurgentes, v. III, Publ. Math. Orsay, 85-5, L'équation du pont et la classification analytique des objects locaux, Université de Paris-Sud, Orsay, 1985 | MR

[9] Howls C.J., “Hyperasymptotics for multidimensional integrals, exact remainder terms and the global connection problem”, Proc. Roy. Soc. London Ser. A, 453 (1997), 2271–2294 | DOI | MR | Zbl

[10] Murphy B.T.M., Wood A.D., “Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank”, Methods Appl. Anal., 4 (1997), 250–260 | DOI | MR | Zbl

[11] Nemes G., Olde Daalhuis A.B., “Asymptotic expansions for the incomplete gamma function in the transition regions”, Math. Comp., 88 (2019), 1805–1827, arXiv: 1803.07841 | DOI | MR | Zbl

[12] Olde Daalhuis A.B., “Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one”, Proc. Roy. Soc. London Ser. A, 454 (1998), 1–29 | DOI | MR | Zbl

[13] Olde Daalhuis A.B., “On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function”, Methods Appl. Anal., 5 (1998), 425–438 | DOI | MR | Zbl

[14] Olde Daalhuis A.B., “On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles”, Methods Appl. Anal., 7 (2000), 727–745 | DOI | MR | Zbl

[15] Olde Daalhuis A.B., “Inverse factorial-series solutions of difference equations”, Proc. Edinb. Math. Soc., 47 (2004), 421–448 | DOI | MR | Zbl

[16] Olde Daalhuis A.B., “Hyperasymptotics for nonlinear ODEs. I A Riccati equation”, Proc. Roy. Soc. London Ser. A, 461 (2005), 2503–2520 | DOI | MR | Zbl

[17] Olde Daalhuis A.B., “Hyperasymptotics for nonlinear ODEs. II The first Painlevé equation and a second-order Riccati equation”, Proc. Roy. Soc. London Ser. A, 461 (2005), 3005–3021 | DOI | MR | Zbl

[18] Olver F.W.J., Olde Daalhuis A.B., Lozier D.W., Schneider B.I., Boisvert R.F., Clark C.W., Miller B.R., Saunders B.V., Cohl H.S., McClain M.A., NIST digital library of mathematical functions, Release 1.1.12 of 2023-12-15, https://dlmf.nist.gov/

[19] Rosser J.B., “Explicit remainder terms for some asymptotic series”, J. Rational Mech. Anal., 4 (1955), 595–626 | DOI | MR | Zbl

[20] Temme N.M., “The asymptotic expansion of the incomplete gamma functions”, SIAM J. Math. Anal., 10 (1979), 757–766 | DOI | MR | Zbl

[21] Temme N.M., Special functions: An introduction to the classical functions of mathematical physics, John Wiley Sons, Inc., New York, 1996 | DOI | MR

[22] Mathematica, Version 14.0, Wolfram Research, Inc., Champaign, IL, 2024