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

@article{Deun2006ASR,
  title={A stable recurrence for the incomplete gamma function with imaginary second argument},
  author={J. V. Deun and R. Cools},
  journal={Numerische Mathematik},
  year={2006},
  volume={104},
  pages={445-456}
}
Even though the two term recurrence relation satisfied by the incomplete gamma function is asymptotically stable in at least one direction, for an imaginary second argument there can be a considerable loss of correct digits before stability sets in. We present 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. A detailed analysis shows that this approach works well… Expand
A Matlab Implementation of an Algorithm for Computing Integrals of Products of Bessel Functions
TLDR
A Matlab program that computes infinite range integrals of an arbitrary product of Bessel functions of the first kind using an integral representation of the upper incomplete Gamma function to integrate the tail of the integrand is presented. Expand
Integrating products of Bessel functions with an additional exponential or rational factor
TLDR
Two Matlab programs are provided to compute integrals of the form ∫ 0 ∞ e − c x x m ∏ i = 1 k J ν i ( a i x ) d x and ∫0 ∞ x m r 2 + x 2 ∏i =1 k Jη i (a i x) with the Bessel function of the first kind and (real) orderν i . Expand
A IIPBF: a MATLAB toolbox for infinite integrals of product of Bessel functions
A MATLAB toolbox, IIPBF, for calculating infinite integrals involving a product of two Bessel functions Ja(ρx)Jb(τx), Ja(ρx)Yb(τx) and Ya(ρx)Yb(τx), for non-negative integers a, b, and a well behavedExpand
Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions
TLDR
Reliability for a broad range of values of ρ and τ for the three different product types as well as different orders in one case is demonstrated. Expand
A Complete Bibliography of Publications in Numerische Mathematik (2020–2029)
acoustic [CWHMB21, ER20]. adapted [GJT20]. Adaptive [EMPS20, FPT20, HPSV21]. ADMM [GSY20]. algebraic [HPSV21]. algorithm [Ber20, HLW20, LO20, NSD20]. algorithms [HJK21, HPSV21, LL21, Loi20]. AmpèreExpand

References

SHOWING 1-10 OF 30 REFERENCES
An asymptotic representation for the Riemann zeta function on the critical line
  • R. Paris
  • 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 isExpand
Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function : (prepublication)
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 asymptoticExpand
Uniform asymptotics for the incomplete gamma functions starting from negative values of the parameters
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 $z sim a$, inExpand
Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral
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 toExpand
The asymptotic expansion of the incomplete gamma functions : (preprint)
Earlier investigations on uniform asymptotic expansions of the incomplete gamma functions are reconsidered. The new results include estimations for the remainder and the extension of the results toExpand
The generalized exponential integral
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 describesExpand
A Matlab Implementation of an Algorithm for Computing Integrals of Products of Bessel Functions
TLDR
A Matlab program that computes infinite range integrals of an arbitrary product of Bessel functions of the first kind using an integral representation of the upper incomplete Gamma function to integrate the tail of the integrand is presented. Expand
Uniform asymptotic smoothing of Stokes’s discontinuities
  • M. Berry
  • 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 itsExpand
Approximation and Computation: A Festschrift in Honor of Walter Gautschi
TLDR
Gauss elimination by segments and multivariate polynomial interpolation, C.C. de Boor quantization of the gamma function, Louis de Branges vector orthogonal polynomials of dimension-d, and a unified approach to recurrence algorithms. Expand
Fredholm integral equations on the Euclidean motion group
In this work, methods for the solution of Fredholm equations of the first kind with convolution kernel are presented, where all the functions in the integral equation are functions on the EuclideanExpand
...
1
2
3
...