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

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 635 ◽  
Author(s):  
Zhen-Hang Yang ◽  
Jing-Feng Tian ◽  
Ya-Ru Zhu
Keyword(s):  
Lower Bound ◽  
Rational Approximation ◽  
Elliptic Integral ◽  
Complete Elliptic Integral ◽  
Lower Approximation

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.

scholarly journals Total cross sections of eγ → $$ eX\overline{X} $$ processes with X = μ, γ, e via multiloop methods

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Roman N. Lee ◽  
Alexey A. Lyubyakin ◽  
Vyacheslav A. Stotsky
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Elliptic Integral ◽  
High Energy ◽  
Calculation Methods ◽  
Complete Elliptic Integral ◽  
Total Cross Sections ◽  
Multiloop Calculation ◽  
The Cross ◽  
Analytical Expressions

Abstract Using modern multiloop calculation methods, we derive the analytical expressions for the total cross sections of the processes e−γ →$$ {e}^{-}X\overline{X} $$ e − X X ¯ with X = μ, γ or e at arbitrary energies. For the first two processes our results are expressed via classical polylogarithms. The cross section of e−γ → e−e−e+ is represented as a one-fold integral of complete elliptic integral K and logarithms. Using our results, we calculate the threshold and high-energy asymptotics and compare them with available results.

scholarly journals Accurate approximations for the complete elliptic integral of the second kind

2016 ◽  
Vol 438 (2) ◽  
pp. 875-888 ◽  
Author(s):  
Zhen-Hang Yang ◽  
Yu-Ming Chu ◽  
Wen Zhang
Keyword(s):  
Elliptic Integral ◽  
Complete Elliptic Integral

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

2018 ◽  
Vol 462 (2) ◽  
pp. 1714-1726 ◽  
Author(s):  
Zhen-Hang Yang ◽  
Wei-Mao Qian ◽  
Yu-Ming Chu ◽  
Wen Zhang
Keyword(s):  
Elliptic Integral ◽  
Geometric Mean ◽  
Complete Elliptic Integral

Special values of the hypergeometric series II

2001 ◽  
Vol 131 (2) ◽  
pp. 309-319 ◽  
Author(s):  
I. J. ZUCKER ◽  
G. S. JOYCE
Keyword(s):  
Gamma Function ◽  
Hypergeometric Series ◽  
Elliptic Integral ◽  
Singular Values ◽  
Complete Elliptic Integral ◽  
Special Values ◽  
Gaussian Hypergeometric Series

Several authors [1, 5, 9] have investigated the algebraic and transcendental values of the Gaussian hypergeometric series(formula here)for rational parameters a, b, c and algebraic and rational values of z ∈ (0, 1). This led to several new identities such as(formula here)and(formula here)where Γ(x) denotes the gamma function. It was pointed out by the present authors [6] that these results, and others like it, could be derived simply by combining certain classical F transformation formulae with the singular values of the complete elliptic integral of the first kind K(k), where k denotes the modulus.Here, we pursue the methods used in [6] to produce further examples of the type (1·2) and (1·3). Thus, we find the following results:(formula here)The result (1·6) is of particular interest because the argument and value of the F function are both rational.

scholarly journals A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus

1970 ◽  
Vol 24 (112) ◽  
pp. 993
Author(s):  
Y. L. L. ◽  
Henry E. Fettis ◽  
James C. Caslin
Keyword(s):  
Elliptic Integral ◽  
Complete Elliptic Integral
Sign in / Sign up

Export Citation Format

Share Document