Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions
| dc.creator | Weniger, Ernst Joachim | |
| dc.date | 2005-11-03 | |
| dc.date.accessioned | 2026-07-07T06:50:43Z | |
| dc.date.available | 2026-07-07T06:50:43Z | |
| dc.description | Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{ν=0}^{\infty} a_ν$ of infinite series $\sum_{ν=0}^{\infty} a_ν$ for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation $Δr_n = a_{n+1}$. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hypergeometric series ${}_2 F_1 (a, b; c; z)$ and the divergent asymptotic inverse power series for the exponential integral $E_1 (z)$ -- the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples. | |
| dc.description | 20 pages, LaTeX2e, 0 figures | |
| dc.identifier | https://arxiv.org/abs/math/0511074 | |
| dc.identifier | http://arxiv.org/abs/math/0511074 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/104750 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 11M06, 33C05, 33C70 | |
| dc.title | Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions | |
| dc.type | text |