Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions
نویسندگان
چکیده
∞ ν=0 aν 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 ∆rn = an+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 2F1(a, b; c; z) and the divergent asymptotic inverse power series for the exponential integral E1(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.
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