# Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters

@article{Temme1996UniformAF, title={Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters}, author={N. Temme}, journal={Methods and applications of analysis}, year={1996}, volume={3}, pages={335-344} }

We consider the asymptotic behavior of the incomplete gamma functions $gamma (-a,-z)$ and $Gamma (-a,-z)$ as $atoinfty$. Uniform expansions are needed to describe the transition area $z sim a$, in which case error functions are used as main approximants. We use integral representations of the incomplete gamma functions and derive a uniform expansion by applying techniques used for the existing uniform expansions for $gamma (a,z)$ and $Gamma (a,z)$. The result is compared with Olver's uniform… Expand

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#### References

SHOWING 1-10 OF 14 REFERENCES

An asymptotic representation for the Riemann zeta function on the critical line

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

A representation for the Riemann zeta function ζ(s) is given as an absolutely convergent expansion involving incomplete gamma functions which is valid for all finite complex values of s (≠ 1). It is… Expand

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

- Mathematics
- 1991

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… Expand

Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function : (prepublication)

- Mathematics
- 1974

New asymptotic expansions are derived for the incomplete gamma functions and the incomplete beta function. In each case the expansion contains the complementary error function and an asymptotic… Expand

Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities

- Mathematics
- Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
- 1996

Uniform asymptotic approximations are derived for the generalized exponential integral Ep(z), where p is real and z complex. Both the cases p → ∞ and |z| → ∞ are considered. For the case p → ∞ an… Expand

Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function

- Mathematics
- Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
- 1996

Uniform asymptotic expansions are derived for solutions of the differential equation d2W/dζ2 = (u2ζ2 + βu + ψ(u, ζ))W, which are uniformly valid for u real and large, β bounded (real or complex), and… Expand

Uniform asymptotic smoothing of Stokes’s discontinuities

- Mathematics
- Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1989

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… Expand

The generalized exponential integral

- Mathematics
- 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… Expand

Special Functions: An Introduction to the Classical Functions of Mathematical Physics

- Mathematics
- 1996

Bernoulli, Euler and Stirling Numbers. Useful Methods and Techniques. The Gamma Function. Differential Equations. Hypergeometric Functions. Orthogonal Polynomials. Confluent Hypergeometric Functions.… Expand

Computation of the incomplete gamma function ratios and their inverse

- Mathematics, Computer Science
- TOMS
- 1986

An algorithm, employing third-order Schröder iteration supported by Newton-Raphson iteration, is provided for computing the incomplete gamma function ratios, using Temme's uniform asymptotic expansions. Expand

Higher Transcendental Functions

- Mathematics
- Nature
- 1955

Higher Transcendental FunctionsBased, in part, on notes left by the late Prof. Harry Bateman, and compiled by the Staff of the Bateman Project. Vol. 1. Pp. xxvi + 302. 52s. Vol. 2. Pp. xvii + 396.… Expand