In our earlier work we developed an algorithm for approximating the locations of discontinuities and the magnitudes of jumps of a bounded function by means of its truncated Fourier series. The algorithm is based on some asymptotic expansion formulas. In the present paper we give proofs for those for
Half integer approximations for the partial sums of the harmonic series
β Scribed by Duane W DeTemple; Shun-Hwa Wang
- Publisher
- Elsevier Science
- Year
- 1991
- Tongue
- English
- Weight
- 311 KB
- Volume
- 160
- Category
- Article
- ISSN
- 0022-247X
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π SIMILAR VOLUMES
By reformulating the equations governing such polygamma functions in terms of a truncated Riemann Zeta function, we derive expressions which enable their evaluation to a much higher degree of accuracy without any additional computational effort.
is non-decreasing. Thus, if one applies the c,-inequality, the inequality follows trivially. From this and from the preceding inequality we obtain By our condition the sums on the right hand side converge to 0 as n -+ + 00. This proves the assertion. Remarks. It is easy to see that for p > 2 our c
When studying the approximation of the wave functions of the \(H\)-atom by sums of Gaussians, Klopper and Kutzelnigg [KK] and Kutzelnigg [Ku] found an asymptotic of \(\exp [-\gamma \sqrt{n}]\). The results were obtained from numerical results and justified by some asymptotic expansions in quadrature