Euler–Maclaurin formula
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1: 24.17 Mathematical Applications
Euler–Maclaurin Summation Formula
…2: 25.2 Definition and Expansions
§25.2(iii) Representations by the Euler–Maclaurin Formula
►3: 2.10 Sums and Sequences
§2.10(i) Euler–Maclaurin Formula
… ►This is the Euler–Maclaurin formula. … ►In both expansions the remainder term is bounded in absolute value by the first neglected term in the sum, and has the same sign, provided that in the case of (2.10.7), truncation takes place at , where is any positive integer satisfying . ►For extensions of the Euler–Maclaurin formula to functions with singularities at or (or both) see Sidi (2004, 2012b, 2012a). … ► …4: Bibliography E
5: 25.11 Hurwitz Zeta Function
§25.11(iii) Representations by the Euler–Maclaurin Formula
►6: Bibliography H
7: Bibliography B
8: 3.5 Quadrature
9: 18.17 Integrals
10: Errata
The title of the paragraph which was previously “Gasper’s -Analog of Clausen’s Formula” has been changed to “Gasper’s -Analog of Clausen’s Formula (16.12.2)”.
§4.13 has been enlarged. The Lambert -function is multi-valued and we use the notation , , for the branches. The original two solutions are identified via and .
Other changes are the introduction of the Wright -function and tree -function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for , additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert -functions in the end of the section.
The wording was changed to make the integration variable more apparent.
Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the icons to the right of each formula, and provide better search capabilities.
It was reported by Nico Temme on 2015-02-28 that the asymptotic formula for is valid for ; originally it was unnecessarily restricted to .