On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. (English) Zbl 0928.34039

Olde Daalhuis, A. B.

Methods Appl. Anal. 5, No. 4, 425-438 (1998).

Summary: The author examines the resurgence properties of the coefficients \(c_r(\eta)\) appearing in a uniform asymptotic expansion of the incomplete gamma function. For the coefficients \(c_r(\eta)\), he gives an asymptotic approximation as \(r\to\infty\) that is a sum of two incomplete beta functions plus a simple asymptotic series in which the coefficients are again \(c_m(\eta)\).
The method is based on the Borel-Laplace transform, which means that next to the asymptotic approximation of \(c_r(\eta)\), one obtains an exponentially-improved asymptotic expansion for the incomplete gamma function.

MSC:

34E05	Asymptotic expansions of solutions to ordinary differential equations
33B20	Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)

Keywords:

incomplete gamma function; Borel-Laplace transform