On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind

By Zhen-Hang Yang, Wei-Mao Qian, Y. Chu, Wen Zhang. Journal of Mathematical Analysis and Applications, 2018

119 citations recorded by the source corpus.

Abstract In the article, we prove that the double inequalities 1 + ( 6 p − 7 ) r ′ p + ( 5 p − 6 ) r ′ π tanh − 1 ⁡ ( r ) 2 r K ( r ) 1 + ( 6 q − 7 ) r ′ q + ( 5 q − 6 ) r ′ π tanh − 1 ⁡ ( r ) 2 r , q A ( 1 , r ) + ( 5 q − 6 ) G ( 1 , r ) A ( 1 , r ) + ( 6 q − 7 ) G ( 1 , r ) L ( 1 , r ) A G M ( 1 , r ) p A ( 1 , r ) + ( 5 p − 6 ) G ( 1 , r ) A ( 1 , r ) + ( 6 p − 7 ) G ( 1 , r ) L ( 1 , r ) hold for all r ∈ ( 0 , 1 ) if and only if p ≥ π / 2 = 1.570796 ⋯ and q ≤ 89 / 69 = 1.289855 ⋯ , where K ( r ) = ∫ 0 π / 2 ( 1 − r 2 sin 2 ⁡ t ) − 1 / 2 d t is the complete elliptic integral of the first kind, tanh − 1 ⁡ ( r ) = log ⁡ [ ( 1 + r ) / ( 1 − r ) ] / 2 is the inverse hyperbolic tangent function, r ′ = 1 − r 2 , and A ( 1 , r ) = ( 1 + r ) / 2 , G ( 1 , r ) = r , L ( 1 , r ) = ( r − 1 ) / log ⁡ r and A G M ( 1 , r ) are the arithmetic, geometric, logarithmic and Gaussian arithmetic-geometric means of 1 and r , respectively.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/968b2d1833c377685d0e44bd6bce76ee76954a4b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

  1. Jingfeng Tian
  2. Zhen-Hang Yang
  3. Y. Chu

In Geometry and Topology

  1. Jingfeng Tian
  2. Zhen-Hang Yang

Citations in the corpus