The Summation of One Class of Infinite Series

• 作者: Jonathan D. Weiss
• 来源期刊: 应用数学(英文)
• 学科: 数学
• 关键词: INFINITE Series HYPERBOLIC FUNCTIONS ELLIPTICAL Nome
• 年，卷（期）: 2014,（17）
• 页码范围: 2815-2822
• 页数: 8页
• 分类号: O1

摘要

This paper presents closed-form expressions for the series, , where the sum is from n = 1 to n = ∞. These expressions were obtained by recasting the series in a different form, followed by the use of certain relationships involving the elliptical nome. Among the values of x for which these expressions can be obtained are of the form: and , where l is an integer between –∞ and ∞. The values of λ include 1,,and 3. Examples of closed-form expressions obtained in this manner are first presented for , , , and . Additional examples are then presented for , , , and . This undertaking was prompted by the author’s work on an electrostatics boundary-value problem related to the van der Pauw measurement technique of electrical resistivity. The presence of this series for x = in the solution of that problem and its absence from any compendium of infinite series that he consulted led to this work.