New Lower Bound for the Generalized Elliptic Integral of the First Kind

Ling Zhu
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Ling Zhu: Department of Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China

Mathematics, 2022, vol. 10, issue 9, 1-13

Abstract: In this paper, we obtain a new simple rational approximation for K a ( r ) : the inequality 2 K a ( r ) / π > g 2 r ′ / g 1 r ′ holds for all r ∈ ( 0 , 1 ) , where K a ( r ) is the generalized elliptic integral of the first kind, r ′ = 1 − r 2 , g 1 r ′ and g 2 r ′ are specific primary and quadratic polynomials about r ′ , respectively. In particular, when a is taken as 1/2, 1/3, 1/4, 1/5 and 1/6 respectively, we can obtain some new specific lower bounds of the corresponding functions.

Keywords: simple bounds; rational approximation; generalized elliptic integral of the first kind (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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