scholarly journals On Alzer and Qiu's Conjecture for Complete Elliptic Integral and Inverse Hyperbolic Tangent Function

2011 ◽  
Vol 2011 ◽  
pp. 1-7 ◽  
Author(s):  
Yu-Ming Chu ◽  
Miao-Kun Wang ◽  
Ye-Fang Qiu
Keyword(s):  
Open Problem ◽  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Hyperbolic Tangent ◽  
Hyperbolic Tangent Function ◽  
Double Inequality ◽  
Tangent Function ◽  
Complete Elliptic Integrals

We prove that the double inequality(π/2)(arthr/r)3/4+α*r<K(r)<(π/2)(arthr/r)3/4+β*rholds for allr∈(0,1)with the best possible constantsα*=0andβ*=1/4, which answer to an open problem proposed by Alzer and Qiu. Here,K(r)is the complete elliptic integrals of the first kind, and arth is the inverse hyperbolic tangent function.

scholarly journals On the Complete Elliptic Integrals and Babylonian Identity I: The 1/Π Formulaes Involving Gamma Functions and Summations

2013 ◽  
Vol 6 ◽  
pp. 40-46
Author(s):  
Edigles Guedes ◽  
Raja Rama Gandhi
Keyword(s):  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Gamma Functions ◽  
Complete Elliptic Integrals

We evaluate the constant 1/Π using the Babylonian identity and complete elliptic integral of first kind. This resulted in two representations in terms of the Euler’s gamma functions and summations.

scholarly journals Answers to three conjectures on convexity of three functions involving complete elliptic integrals of the first kind

2020 ◽  
Vol 14 (1) ◽  
pp. 255-271 ◽  
Author(s):  
Miao-Kun Wang ◽  
Hong-Hu Chu ◽  
Yong-Min Li ◽  
Yu-Ming Chu
Keyword(s):  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Complete Elliptic Integrals ◽  
Absolutely Monotonic

In the article, we prove that the function x ? (1-x)pK(?x) is logarithmically concave on (0,1) if and only if p ? 7/32, the function x ? K(?x)/log(1+4/?1-x) is convex on (0,1) and the function x ? d2/dx2 [K(?x)- log (1+4/?1-x) is absolutely monotonic on (0,1), where K(x) = ??/20 (1-x2 sin2t)-1/2 dt (0 < x < 1) is the complete elliptic integral of the first kind.

scholarly journals On the Complete Elliptic Integrals and Babylonian Identity II: An Approximation for the Complete Elliptic Integral of the First Kind

2013 ◽  
Vol 6 ◽  
pp. 47-54
Author(s):  
Edigles Guedes ◽  
Raja Rama Gandhi
Keyword(s):  
Closed Form ◽  
Bessel Functions ◽  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Complete Elliptic Integrals

Using the Theorem 4 of previous paper, I evaluate the complete elliptic integral of the first kind in approximate analytical closed form, by means of Bessel functions.

scholarly journals On the Complete Elliptic Integrals and Babylonian Identity III: An Approximation for the Complete Elliptic Integral of the First Kind

2013 ◽  
Vol 6 ◽  
pp. 55-59
Author(s):  
Edigles Guedes ◽  
Raja Rama Gandhi
Keyword(s):  
Closed Form ◽  
Bessel Functions ◽  
Elliptic Integral ◽  
Elliptic Integrals ◽  
Complete Elliptic Integral ◽  
Complete Elliptic Integrals

Using the Corollary 3 of previous paper (http://www.bmsa.us/admin/uploads/Mn95aZ.pdf), we evaluate the complete elliptic integral of the first kind in an approximately equal analytical closed form, by means of Bessel functions.

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