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

@inproceedings{Temme1974UniformAE, title={Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function : (prepublication)}, author={Nico M. Temme}, year={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 series. The expansions are uniformly valid with respect to certain domains of the parameters.

#### 78 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

Products of Incomplete gamma functions Integral representations

- Mathematics
- 2016

In this paper we find integral representations, involving incomplete gamma and incomplete beta functions, of products of incomplete gamma functions. Also, in this paper we find interesting relations… Expand

A numerical study of a new asymptotic expansion for the incomplete gamma function

- 2004

In this paper we develop further an asymptotic expansion recently derived by Kowalenko and FVankel [6] for a particular Rummer function that is related to the incomplete gamma function. This… Expand

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

- Mathematics
- 1996

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… Expand

More Properties of the Incomplete Gamma Functions

- Mathematics
- 2014

In this paper, additional properties of the lower gamma functions and the error functions are introduced and proven. In particular, we prove interesting relations between the error functions and… Expand

A uniform asymptotic expansion for the jacobi polynomials with explicit remainder

- Mathematics
- 1996

A new asymptotic expansion is derived for the Jacobi polynonmial which holds uniformly for .An explicit expression is also given for the error term associated with the expansion. Our approach begins… Expand

ANTIDERIVATIVES AND INTEGRALS INVOLVING INCOMPLETE BETA FUNCTIONS WITH APPLICATIONS

- 2020

In this paper, we prove that incomplete beta functions are antiderivatives of several products and powers of trigonometric functions, we give formulas for antiderivatives for products and powers of… Expand

Products of incomplete gamma functions

- Mathematics
- 2015

Abstract Many properties of gamma functions are known. In this paper, we extend similar properties to incomplete gamma functions. In particular, it is known that Γ ( 1 - a ) Γ ( a ) = π csc … Expand

A stable recurrence for the incomplete gamma function with imaginary second argument

- Mathematics, Computer Science
- Numerische Mathematik
- 2006

This work presents an approach to compute the recurrence relation to full precision, also for small values of the arguments, when the first argument is negative and the second one is purely imaginary. Expand

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

- Mathematics, Computer Science
- Numerical Algorithms
- 2004

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… Expand

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Some asymptotic formulae given elsewhere for the zeros of the incomplete gam- ma function -y(a, x) are corrected. A plot of a few of the zero trajectories of the function y(xw, x) is given, where x… Expand

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A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.

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@BULLET {(1 -x0)/(x0 -x) + tj-^KI + OOT1)), q -* ~

- @BULLET {(1 -x0)/(x0 -x) + tj-^KI + OOT1)), q -* ~

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