Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1117.65042
Weniger, Ernst Joachim
Asymptotic approximations to truncation errors of series representations for special functions. (English)
[A] Iske, Armin (ed) et al., Algorithms for approximation. Proceedings of the 5th international conference, Chester, UK, July 17--21, 2005. Berlin: Springer. 331-348 (2007). ISBN 3-540-33283-9/hbk

Summary: Asymptotic approximations $(n\to\infty)$ to the truncation errors $r_n=-\sum^\infty_{\nu=n+1}a_\nu$ of infinite series $\sum^\infty_{\nu=0}a_\nu$ 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 $\Delta r_n=a_{n+1}$. 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 $_2F_1(a,b;c;z)$ 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.

MSC 2000:: *65D20 Computation of special functions
33F05 Numerical approximation of special functions
11M06 Riemannian zeta-function and Dirichlet L-function
33C20 Generalized hypergeometric series
33E20 Functions defined by series and integrals

Keywords: symbolic computation; Asymptotic approximations; special functions; Dirichlet series; Riemann zeta function; Euler-Maclaurin formula; Gaussian hypergeometric series; asymptotic inverse power series; exponential integral; numerical examples

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]