A Rational Approximation for the Complete Elliptic Integral of the First Kind

Zhen-Hang Yang, Jing-Feng Tian and Ya-Ru Zhu
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Zhen-Hang Yang: Engineering Research Center of Intelligent Computing for Complex Energy Systems of Ministry of Education, North China Electric Power University, Yonghua Street 619, Baoding 071003, China
Jing-Feng Tian: Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, Baoding 071003, China
Ya-Ru Zhu: Department of Mathematics and Physics, North China Electric Power University, Yonghua Street 619, Baoding 071003, China

Mathematics, 2020, vol. 8, issue 4, 1-9

Abstract: Let K ( r ) be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for K ( r ) . More precisely, we establish the inequality 2 π K ( r ) > 5 ( r ′ ) 2 + 126 r ′ + 61 61 ( r ′ ) 2 + 110 r ′ + 21 for r ∈ ( 0 , 1 ) , where r ′ = 1 − r 2 . The lower bound is sharp.

Keywords: complete integrals of the first kind; arithmetic-geometric mean; rational approximation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

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