# Asymptotic and exact series representations for the incomplete Gamma function

@article{Amore2005AsymptoticAE, title={Asymptotic and exact series representations for the incomplete Gamma function}, author={P. Amore}, journal={EPL}, year={2005}, volume={71}, pages={1-7} }

Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly convergent series, completely analytical, which can be used to obtain arbitrarily accurate estimates of Γ(a,x) for any value of a or x. Applications of these formulas are discussed.

#### 36 Citations

Convergent expansions of the incomplete gamma functions in terms of elementary functions

- Mathematics
- 2017

We consider the incomplete gamma function γ(a,z) for ℜa > 0 and z ∈ ℂ. We derive several convergent expansions of z−aγ(a,z) in terms of exponentials and rational functions of z that hold uniformly in… Expand

Series Expansion of Generalized Fresnel Integrals

- Mathematics
- 2012

The two Fresnel Integrals are real and imaginary part of the integral over complex-valued exp(ix^2) as a function of the upper limit. They are special cases of the integrals over x^m*exp(i*x^n) for… Expand

Analytic parametric equations of log-aesthetic curves in terms of incomplete gamma functions

- Mathematics, Computer Science
- Comput. Aided Geom. Des.
- 2012

Log-aesthetic curves (LACs) have recently been developed to meet the requirements of industrial design for visually pleasing shapes. LACs are defined in terms of definite integrals, and adaptive… Expand

A note on Love's equation with damping

- Mathematics
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2006

The dynamic propagation of a Heaviside input signal in a semi-infinite, dissipative/dispersive medium is considered. The exact solution to this problem, which corresponds to Stokes' first problem of… Expand

Slepian Spatial-Spectral Concentration on the Ball

- Mathematics, Computer Science
- ArXiv
- 2014

The Slepian spatial and spectral concentration problems are formulated as eigenvalue problems, the eigenfunctions of which form an orthogonal family of concentrated functions that form a SlePian basis that permits a sparse representation of concentrated signals efficiently. Expand

Multivariate extremal density expansions and residual tail dependence structures

- Chemistry
- 2009

In multivariate extreme value theory dependence structures can be modeled by using Pickands dependence functions. Extreme value distribution functions (EVDs) with standard reversely exponential… Expand

Hydrodynamic description of the long-time tails of the linear and rotational velocity autocorrelation functions of a particle in a confined geometry.

- Physics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2007

A hydrodynamic analysis of the long-time behavior of the linear and angular velocity autocorrelation functions of an isolated colloid particle constrained to have quasi-two-dimensional motion and uses the singularity method to characterize unsteady linear motion of an incompressible fluid. Expand

An Improved Analytical Solution for the Temperature Profile of Ice Sheets

- Geology
- 2019

The one-dimensional steady state analytical solution of the energy conservation equation obtained by Robin (1955, https://doi.org/10.3189/002214355793702028) is frequently used in glaciology. This… Expand

Shock and traveling wave phenomena on an externally damped, non-linear string

- Mathematics
- 2009

Abstract We examine the propagation of shocks and traveling wave phenomena on a one-dimensional string that is executing finite-amplitude, transverse vibrations in a resisting medium. As part of our… Expand

INVESTIGATION OF THE ENTROPY AND SPECIFIC HEAT CAPACITY OF GaN USING INCOMPLETE GAMMA FUNCTIONS

- Materials Science
- 2008

The aim of this paper is to provide validity and reliable analytical relation for the thermodynamic functions calculated in terms of the Debye temperature using incomplete gamma functions, and… Expand

#### References

SHOWING 1-10 OF 16 REFERENCES

A method for classical and quantum mechanics

- Physics, Mathematics
- 2004

In many physical problems it is not possible to find an exact solution. However, when some parameter in the problem is small, one can obtain an approximate solution by expanding in this parameter.… Expand

CALCULATING NORMAL PROBABILITIES

- Mathematics
- 1995

This is probably the best known example of an integral that cannot be evaluated in terms of elementary functions. In this note we develop an elementary approximation to P(a) which arises in a natural… Expand

Large Order Perturbation Theory and Summation Methods in Quantum Mechanics

- Mathematics
- 1990

The problems usually found in Theoretical Chemistry and Physical Chemistry involve the use of quantum mechanical models which, as a general rule, do not have exact solutions. Due to this very… Expand

Path Integrals in Quantum Mechanics Statistics and Polymer Physics

- Mathematics
- 1990

Elementary properties and simple solutions external sources, correlation function and perturbation theory semiclassical time displacement amplitude Feynman-Kleinert variational approach and… Expand

Phys

- Rev. D 23, 2916
- 1981

Phys

- Rev. D 63, 125013
- 2001

Phys. Rev. D

- Phys. Rev. D
- 2001

The incomplete gamma functions since Tricomi

- “Tricomi’s Ideas and Contemporary Applied Mathematics”, Atti Convegni Lincei, Vol. 147, Accad. Naz. Lincei, Rome
- 1998

Calculating normal probabilities, The American Mathematical Montly

- Calculating normal probabilities, The American Mathematical Montly
- 1995