scholarly journals Error Bounds for the Large-Argument Asymptotic Expansions of the Lommel and Allied Functions

2018 ◽  
Vol 140 (4) ◽  
pp. 508-541 ◽  
Author(s):  
Gergő Nemes
Keyword(s):  
Error Bounds ◽  
Asymptotic Expansions

scholarly journals Uniform asymptotic expansions for the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) with μ large

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210360
Author(s):  
T. M. Dunster
Keyword(s):  
Analytic Continuation ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Whittaker Functions ◽  
Airy Functions ◽  
Elementary Functions ◽  
Confluent Hypergeometric Functions ◽  
Uniform Asymptotic Expansions

Uniform asymptotic expansions are derived for Whittaker’s confluent hypergeometric functions M κ , μ ( z ) and W κ , μ ( z ) , as well as the numerically satisfactory companion function W − κ , μ ( z   e − π i ) . The expansions are uniformly valid for μ → ∞ , 0 ≤ κ / μ ≤ 1 − δ < 1 and 0 ≤ arg ⁡ ( z ) ≤ π . By using appropriate connection and analytic continuation formulae, these expansions can be extended to all unbounded non-zero complex z . The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.

scholarly journals Simplified error bounds for turning point expansions

2020 ◽  
pp. 1-32
Author(s):  
T. M. Dunster ◽  
A. Gil ◽  
J. Segura
Keyword(s):  
Differential Equations ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Turning Point ◽  
Linear Differential Equations ◽  
Airy Functions ◽  
Elementary Functions ◽  
Varying Coefficient ◽  
Linear Differential ◽  
High Degree

Recently, the present authors derived new asymptotic expansions for linear differential equations having a simple turning point. These involve Airy functions and slowly varying coefficient functions, and were simpler than previous approximations, in particular being computable to a high degree of accuracy. Here we present explicit error bounds for these expansions which only involve elementary functions, and thereby provide a simplification of the bounds associated with the classical expansions of Olver.

scholarly journals Error bounds for the asymptotic expansion of the Hurwitz zeta function

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170363 ◽  
Author(s):  
G. Nemes
Keyword(s):  
Asymptotic Expansion ◽  
Zeta Function ◽  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Remainder Term ◽  
Hurwitz Zeta Function ◽  
Polygamma Functions

In this paper, we reconsider the large- a asymptotic expansion of the Hurwitz zeta function ζ ( s , a ). New representations for the remainder term of the asymptotic expansion are found and used to obtain sharp and realistic error bounds. Applications to the asymptotic expansions of the polygamma functions, the gamma function, the Barnes G -function and the s -derivative of the Hurwitz zeta function ζ ( s , a ) are provided. A detailed discussion on the sharpness of our error bounds is also given.

scholarly journals The resurgence properties of the large-order asymptotics of the Hankel and Bessel functions

2014 ◽  
Vol 12 (04) ◽  
pp. 403-462 ◽  
Author(s):  
Gergő Nemes
Keyword(s):  
Error Bounds ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Smooth Transition ◽  
Large Order ◽  
Computable Error Bounds

The aim of this paper is to derive new representations for the Hankel and Bessel functions, exploiting the reformulation of the method of steepest descents by Berry and Howls [Hyperasymptotics for integrals with saddles, Proc. R. Soc. Lond. A 434 (1991) 657–675]. Using these representations, we obtain a number of properties of the large-order asymptotic expansions of the Hankel and Bessel functions due to Debye, including explicit and numerically computable error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.

scholarly journals Liouville-Green expansions of exponential form, with an application to modified Bessel functions

2019 ◽  
Vol 150 (3) ◽  
pp. 1289-1311 ◽  
Author(s):  
T. M. Dunster
Keyword(s):  
Error Bounds ◽  
Asymptotic Expansions ◽  
Turning Point ◽  
Bessel Functions ◽  
Alternative Form ◽  
Modified Bessel Functions ◽  
Exponential Form ◽  
Positive Order ◽  
Second Order Differential Equations ◽  
Computable Error Bounds

AbstractLinear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z).

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