representations by Euler–Maclaurin formula
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§25.2(iii) Representations by the Euler–Maclaurin Formula…
§25.11(iii) Representations by the Euler–Maclaurin Formula…
§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 representation in terms of was added to this equation.
The wording was changed to make the integration variable more apparent.
The Olver hypergeometric function , previously omitted from the left-hand sides to make the formulas more concise, has been added. In Equations (15.6.1)–(15.6.5), (15.6.7)–(15.6.9), the constraint has been added. In (15.6.6), the constraint has been added. In Section 15.6 Integral Representations, the sentence immediately following (15.6.9), “These representations are valid when , except (15.6.6) which holds for .”, has been removed.
It was reported by Nico Temme on 2015-02-28 that the asymptotic formula for is valid for ; originally it was unnecessarily restricted to .