# Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument

@article{Mathar2004NumericalRO, title={Numerical Representations of the Incomplete Gamma Function of Complex-Valued Argument}, author={R. Mathar}, journal={Numerical Algorithms}, year={2004}, volume={36}, pages={247-264} }

Various approaches to the numerical representation of the incomplete Gamma function γ(m+1/2,z) for complex arguments z and non-negative small integer indices m are compared with respect to numerical fitness (accuracy and speed). We consider power series, Laurent series, classical numerical methods of sampling the basic integral representation, and others not yet covered by the literature. The most suitable scheme is the construction of Taylor expansions around nodes of a regular, fixed grid in… Expand

#### Figures, Tables, and Topics from this paper

#### 9 Citations

Numerical methods for the computation of the confluent and Gauss hypergeometric functions

- Computer Science, Mathematics
- Numerical Algorithms
- 2016

The available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes are reviewed and recommendations for which methods should be used in each situation are provided. Expand

A fast algorithm for computing the Boys function

- Computer Science, Mathematics
- ArXiv
- 2021

The Boys function is related to a number of special functions, for example the error function, the incomplete Gamma function as well as (for pure imaginary argument) to the Fresnel integrals. Expand

Computation of Hypergeometric Functions

- Mathematics
- 2009

We seek accurate, fast and reliable computations of the confluent and Gauss hypergeometric functions 1F1(a; b; z) and 2F1(a, b; c; z) for different parameter regimes within the complex plane for the… Expand

Third Order Newton's Method for Zernike Polynomial Zeros

- Mathematics
- 2007

The Zernike radial polynomials are a system of orthogonal polynomials over the unit interval with weight x. They are used as basis functions in optics to expand fields over the cross section of… Expand

Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

- Chemistry
- 2010

Abstract In this article we combine the favorable properties of efficient Gaussian type orbitals basis sets, which are applied with good success in conventional electronic structure methods, and… Expand

Efficient Calculation of Molecular Integrals over London Atomic Orbitals.

- Physics, Medicine
- Journal of chemical theory and computation
- 2017

This work presents an implementation of several different algorithms for the evaluation of ERIs over Gaussian-type LAOs at arbitrary magnetic field strengths, and selects the most efficient approach to calculate the ERIs for each shell quartet. Expand

Optimizing Molecular Geometries in Strong Magnetic Fields

- Medicine
- Journal of chemical theory and computation
- 2021

The present implementation of molecular gradients at the CDFT level provides a cost-effective approach to the study of molecular structure under strong magnetic fields, opening up many new possibilities for the studyof chemistry in this regime. Expand

Application of the Algebraic Difference Approach for Developing Self-Referencing Specific Gravity and Biomass Equations

- Mathematics
- 2006

Biomass estimation is critical for looking at ecosystem processes and as a measure of stand yield. The density-integral approach allows for coincident estimation of stem profile and biomass. The… Expand

Outage performance of cognitive two-way amplify-and-forward relay network under different transmission schemes

- Computer Science
- Trans. Emerg. Telecommun. Technol.
- 2020

#### References

SHOWING 1-10 OF 37 REFERENCES

On the computation of incomplete gamma functions in the complex domain

- Mathematics
- 1985

Abstract Some new continued fractions for incomplete gamma functions γ( a, z ) and Γ( a, z ), with a and z complex, are derived. For many of these expansions, it is shown that the approximants can be… Expand

A uniform asymptotic expansion for the incomplete gamma function

- Mathematics
- 2002

We describe a new uniform asymptotic expansion for the incomplete gamma function Γ(a,z) valid for large values of z. This expansion contains a complementary error function of an argument measuring… Expand

A continued fraction algorithm for the computation of higher transcendental functions in the complex plane

- Mathematics
- 1967

This report deals with the numerical evaluation of a class of func- tions of a complex variable that can be represented as Stieltjes transforms of non- negative real functions. The considered class… Expand

Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials

- Mathematics
- 1961

to generate the necessary formulas. An alternate procedure utilizes the Lommel functions of two variables [2]; unfortunately, the Lommel functions have not been extensively tabulated. The remaining… Expand

Numerical integration using rys polynomials

- Mathematics
- 1976

Abstract We define and discuss the properties of manifolds of polynomials J n ( t , x ) and R n ( t , x ), called Rys polynomials, which are orthonormal with respect to the weighting factor exp(− xt… Expand

The rational approximation of functions which are formally defined by a power series expansion

- Mathematics
- 1960

1. Introduction. The advent of high speed digital computers and the consequent intensification of interest in the study of numerical analysis has caused considerable attention to be paid to the… Expand

The asymptotic expansion of integral functions defined by Taylor series

- Mathematics
- Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
- 1940

1.1. Considerable attention has been devoted to the behaviour of the general integral function for large values of the variable, and many important theorems have been proved in this field. On the… Expand

The Asymptotic Expansion of Integral Functions Defined by Taylor's Series

- Mathematics
- 1906

1. Integral functions can be defined either by Taylor’s series or Weierstrassian products. When the zeros are simple functions of their order number, the latter method is, as a rule, most simple.… Expand

Incomplete GammaFm(x) Functions for Real Negative and Complex Arguments

- Mathematics
- 1998

Incomplete gamma functionsFm(x), originally defined and used in the electronic structure theory, have been examined from the viewpoint of electron?molecule scattering theory for their possible use in… 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