Functional Inequalities for Complete Elliptic Integrals and Their Ratios

Asymptotic formulas with error bounds are obtained for the first elliptic integral near its logarithmic singularity. It is convenient to start from the more general problem of expanding the integral over the positive t-axis of $[(t + x)(t + y)(t + z)(t + w)]^{{{ - 1} / 2}} $. The method of Mellin transforms gives an asymptotic expansion that converges uniformly if ${{0 < \max \{ x,y\} } / {\min \{ z,w\} \leqq r < 1}}$. Each term of the series contains a Legendre polynomial and the derivative of a Legendre function with respect to its degree; this derivative involves nothing worse than a logarithm. A simple bound for the relative error of the Nth partial sum is obtained from Wong’s formula for the remainder, aided by Chebyshev’s integral inequality. Error bounds are given also for more accurate asymptotic formulas containing a complete elliptic integral. Formulas for the standard elliptic integral of the first kind are obtained in the case $w = \infty $. The method of Mellin transforms and Wong’s formula are discussed in an appendix.

Journal Article Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind Get access J. M. BORWEIN, J. M. BORWEIN Mathematics, Statistics and Computing Department, Dalhousie UniversityHalifax, Nova Scotia B3H 3J5 Search for other works by this author on: Oxford Academic Google Scholar I. J. ZUCKER I. J. ZUCKER Physics Department, King's CollegeLondon, WC2R 2LS Search for other works by this author on: Oxford Academic Google Scholar IMA Journal of Numerical Analysis, Volume 12, Issue 4, October 1992, Pages 519–526, https://doi.org/10.1093/imanum/12.4.519 Published: 01 October 1992 Article history Received: 30 July 1990 Revision received: 30 January 1991 Published: 01 October 1992

In this paper, we prove that Hp(K (r),K (r′)) K ( √ 2/2) and Hq(K (r),K (r′)) K ( √ 2/2) for all r ∈ (0,1) if and only if p 1− 4[K (√2/2)]4/π2 = −3.789 · · · and q (log2)/[log(π/2)− logK (√2/2)] = −4.1805 · · · , where Hp(x,y) denotes the Holder mean of order p of two positive numbers x and y , r′ = √ 1− r2 , and K (r) denotes the complete elliptic integral of the first kind, respectively. Mathematics subject classification (2010): Primary 26E60.

The contour integrals, occurring in the arbitrary-order phase-integral quantization conditions given in a previous paper, are in the first- and third-order approximations expressed in terms of complete elliptic integrals in the case that the charges of the Coulomb centers are equal. The evaluation of the integrals is facilitated by the knowledge of quasiclassical dynamics. The resulting quantization conditions involving complete elliptic integrals are solved numerically to obtain the energy eigenvalues and the separation constants of the 1sσ and 2pσ states of the hydrogen molecule ion for various values of the internuclear distance. The accuracy of the formulas obtained is illustrated by comparison with available numerically exact results.

Generalized trigonometric functions are applied to the Legendre-Jacobi standard form of complete elliptic integrals, and a new form of the generalized complete elliptic integrals of the Borweins is presented. According to the form, it can be easily shown that these integrals have similar properties to the classical ones. In particular, it is possible to establish a computation formula of the generalized $\pi$ in terms of the arithmetic-geometric mean, in the classical way as the Gauss-Legendre algorithm for $\pi$ by Salamin and Brent. Moreover, an elementary new proof of Ramanujan's cubic transformation is also given.

Let K(k) and E(k) denote respectively the complete elliptic integrals of the first and second kind with modulus k, as defined by Byrd and Friedman ([5] 110·06 and 110·07), and let k′ = √(1 − k2), the complementary modulus.

Legendre hyperelliptic integrals, π new formulae and Lauricella functions through the elliptic singular moduli

6.-7 - DEFINITE INTEGRALS OF SPECIAL FUNCTIONS

Two sets of closed analytic functions are proposed for the approximate calculus of the complete elliptic integrals of the 1st and 2nd kinds in the normal form due to Legendre, their expressions having a remarkable simplicity and accuracy. The special usefulness of the newly proposed formulas consists in that they allow performing the analytic study of variation of the functions in which they appear, using the derivatives (they being expressed in terms of elementary functions only, without any special function; this would mean replacing one difficulty by another of the same kind). Comparative tables of the approximate values so obtained and the exact ones, reproduced from special functions tables are given (vs. the elliptic integrals modulus k). It is to be noticed that both sets of formulas are given neither by spline nor by regression functions, but by asymptotic expansions, the identity with the exact functions being accomplished for the left domain’s end. As for their simplicity, the formulas in k / k' do not need any mathematical table (are purely algebraic). As for their accuracy, the 2nd set, although more intricate, gives more accurate values than the 1st one and extends itself more closely to the right domain’s end.

In the paper, the authors present some monotonicity properties of certain functions defined in terms of the complete elliptic integrals of the second kind and some elementary functions and, consequently, improve several known inequalities for the complete elliptic integrals of the second kind.

Sharp Landen transformation inequalities for hypergeometric functions, with applications

A particular lattice Green function for the face centred cubic lattice with anisotropic nearest neighbour interactions is evaluated exactly as a product of two complete elliptic integrals. Applications of this result in lattice statistics are briefly described.

ABSTRACTAn earlier method for obtaining indefinite integrals of special function from the second-order linear equations which define them has been reformulated in terms of Riccati equations, which are nonlinear and first-order. For this application the nonlinearity is not a problem and the first-order property is a great advantage. Integrals can be derived using fragments of these Riccati equations and here only two specific fragment types are examined in detail. These fragments allow general integration formulas to be derived using quadrature. Other results will be presented separately. Results are presented here for Airy functions, Bessel functions, complete elliptic integrals, associated Legendre functions and Gauss hypergeometric functions. All results have been checked by differentiation using Mathematica.

We study a family of the Laurent biorthogonal polynomials arising from the Hermite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients, explicit expression and the weight function for these polynomials are obtained. We construct also a new explicit example of the Szeg\"o polynomials orthogonal on the unit circle. Relations with associated Legendre polynomials are considered.

Analytical and numerical methods for massive two-loop self-energy diagrams