# Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral

@article{Olver1991UniformEI, title={Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral}, author={Frank W. J. Olver}, journal={Siam Journal on Mathematical Analysis}, year={1991}, volume={22}, pages={1475-1489} }

By allowing the number of terms in an asymptotic expansion to depend on the asymptotic variable, it is possible to obtain an error term that is exponentially small as the asymptotic variable tends to its limit. This procedure is called “exponential improvement.” It is shown how to improve exponentially the well-known Poincare expansions for the generalized exponential integral (or incomplete Gamma function) of large argument. New uniform expansions are derived in terms of elementary functions…

## 68 Citations

Error bounds and exponential improvement for the asymptotic expansion of the Barnes G-function

- MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2014

In this paper, we establish new integral representations for the remainder term of the known asymptotic expansion of the logarithm of the Barnes G-function. Using these representations, we obtain…

Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions

- Mathematics
- 2016

In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms…

The generalized exponential integral

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- 1994

This paper concerns the role of the generalized exponential integral in recently-developed theories of exponentially-improved asymptotic expansions and the Stokes phenomenon. The first part describes…

Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2015

In 1994 Boyd derived a resurgence representation for the gamma function, exploiting the 1991 reformulation of the method of steepest descents by Berry and Howls. Using this representation, he was…

ERROR BOUNDS AND EXPONENTIAL IMPROVEMENT FOR HERMITE'S ASYMPTOTIC EXPANSION FOR THE GAMMA FUNCTION

- Mathematics, Philosophy
- 2013

In this paper we reconsider the asymptotic expansion of the Gamma function
with shifted argument, which is the generalization of the well-known
Stirling series. To our knowledge, no explicit…

Uniform asymptotic expansions of integrals: a selection of problems

- Mathematics, Computer Science
- 1995

Exponentially improved asymptotic solutions of ordinary differential equations. II. Irregular singularities of rank one

- MathematicsProceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
- 1994

Re-expansions are found for the optimal remainder terms in the well-known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an…

Exponentially small expansions related to the parabolic cylinder function.

- Mathematics
- 2019

The refined asymptotic expansion of the confluent hypergeometric function $M(a,b,z)$ on the Stokes line $\arg\,z=\pi$ given in {\it Appl. Math. Sci.} {\bf 7} (2013) 6601--6609 is employed to derive…

UNIFORM ASYMPTOTICS FOR THE INCOMPLETE GAMMA FUNCTIONS STARTING FROM NEGATIVE VALUES OF THE PARAMETERS

- Mathematics
- 2005

. We consider the asymptotic behavior of the incomplete gamma func tions 1'(-a,-z) and I'(-a,-z) as a-+ oo. Uniform expansions are needed to describe the transition area z ,..., a, in which case…

Exponentially small expansions of the Wright function on the Stokes lines

- Mathematics
- 2014

We investigate a particular aspect of the asymptotic expansion of the Wright function pΨq(z) for large |z|. In the case p = 1, q ⩾ 0, we establish the form of the exponentially small expansion of…

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Across a Stokes line, where one exponential in an asymptotic expansion maximally dominates another, the multiplier of the small exponential changes rapidly. If the expansion is truncated near its…