Very accurate approximations for the elliptic integrals of the second kind in terms of Stolarsky means

2015/08/22 by Zhen-Hang Yang, Yang, Zhen-Hang
Mathematics · #26A48 #26D15 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Primary 33E05 #Secondary 26E60 #math.CA #msc:26A48 #msc:26D15 #msc:26E60 #msc:33E05

paper · pdf · doi:10.48550/arxiv.1508.05513

25 pages

arxiv created 2015/08/22 · openalex publication_date 2015/08/22 · arxiv updated 2015/08/25 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

For a,b>0 with a≠ b, the Stolarsky means are defined by% S_p,q(a,b) =(\dfracq(a^p-b^p)p(a^q-b^q)% ) ^1/(p-q)ifpq(p-q) ≠ 0% and S_p,q(a,b) is defined as its limits at p=0 or q=0 or p=q if pq(p-q) =0. The complete elliptic integrals of the second kind E is defined on (0,1) by% E(r) =∫₀^π/2√1-r²sin ²tdt.% We prove that the functions% F(r) =\frac1-(2/π) E(r)% 1-S_11/4,7/4(1,r^′)andG(r) =% \frac1-(2/π) E(r)1-S_5/2,2(1,r^′)% are strictly decreasing and increasing on (0,1) , respectively, where r^′=√1-r². These yield some very accurate approximations for the complete elliptic integrals of the second kind, which greatly improve some known results.

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