Properties and Applications of the Reciprocal Logarithm Numbers View Full Text


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Article Info

DATE

2010-02

AUTHORS

Victor Kowalenko

ABSTRACT

Via a graphical method, which codes tree diagrams composed of partitions, a novel power series expansion is derived for the reciprocal of the logarithmic function ln (1+z), whose coefficients represent an infinite set of fractions. These numbers, which are called reciprocal logarithm numbers and are denoted by Ak, converge to zero as k→∞. Several properties of the numbers are then obtained including recursion relations and their relationship with the Stirling numbers of the first kind. Also appearing here are several applications including a new representation for Euler’s constant known as Hurst’s formula and another for the logarithmic integral. From the properties of the Ak it is found that a term of ζ(2) cannot be eliminated by the remaining terms in Hurst’s formula, thereby indicating that Euler’s constant is irrational. Finally, another power series expansion for the reciprocal of arctangent is developed by adapting the preceding material. More... »

PAGES

413-437

References to SciGraph publications

Journal

TITLE

Acta Applicandae Mathematicae

ISSUE

2

VOLUME

109

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s10440-008-9325-0

DOI

http://dx.doi.org/10.1007/s10440-008-9325-0

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1011200235


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