5,865 research outputs found
Error bounds and exponential improvement for the asymptotic expansion of the Barnes -function
In this paper we establish new integral representations for the remainder
term of the known asymptotic expansion of the logarithm of the Barnes
-function. Using these representations, we obtain explicit and numerically
computable error bounds for the asymptotic series, which are much simpler than
the ones obtained earlier by other authors. We find that along the imaginary
axis, suddenly infinitely many exponentially small terms appear in the
asymptotic expansion of the Barnes -function. Employing one of our
representations for the remainder term, we derive an exponentially improved
asymptotic expansion for the logarithm of the Barnes -function, which shows
that the appearance of these exponentially small terms is in fact smooth,
thereby proving the Berry transition property of the asymptotic series of the
-function.Comment: 14 pages, accepted for publication in Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Science
The resurgence properties of the incomplete gamma function I
In this paper we derive new representations for the incomplete gamma
function, exploiting the reformulation of the method of steepest descents by C.
J. Howls (Howls, Proc. R. Soc. Lond. A 439 (1992) 373--396). Using these
representations, we obtain a number of properties of the asymptotic expansions
of the incomplete gamma function with large arguments, including explicit and
realistic error bounds, asymptotics for the late coefficients, exponentially
improved asymptotic expansions, and the smooth transition of the Stokes
discontinuities.Comment: 36 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1311.2522, arXiv:1309.2209, arXiv:1312.276
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
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