Partial Fraction Expansion of Meromorphic Maps

Abstract

Partial fraction expansion of a meromorphic function or map is a way of expressing it as an infinite series of rational functions and polynomials. This representation is particularly useful in the summation of series in mathematical analysis and has at its heart the Mittag-Leffler’s expansion theorem which holds for arbitrary domains. In this article, the Mittag-Leffler’s expansion theorem was used to obtain partial fraction expansions of the functions tan⁡z and coth⁡z which are meromorphic in the entire complex plane. The expansion of tan⁡z was then used to obtain the Laurent series for it where except for the odd coefficients, all the even coefficients in the series were shown to vanish identically. By comparing these coefficients with those of the well-known Maclaurin series for tan⁡z the exact sum of the odd terms in the important p-series ∑_(n=1)^∞▒1/n^p (p=2,4,6) were obtained in closed form and consequently, those of the series ∑_(n=1)^∞▒1/n^p (p=2,4,6) for all values of n. The partial fraction expansion of coth⁡z was also gainfully employed in finding the exact sum (in closed form) of the p-series ∑_(n=1)^∞▒1/n^2 and ∑_(n=-∞)^∞▒〖1/(n^2+z^2 ) (z≠ni,n=0,1,2,…)〗 to also show its usefulness in application. Finally, since closed form expressions were obtained for those series found from the partial fraction expansions of the functions considered in this article, we suggest that further research in this area should focus on the application of the Mittag-Leffler’s theorem to other meromorphic functions such as sec⁡z, cot⁡z etc to find similar results on the summation of series as done here.