Numerical calculation of elliptic integrals and elliptic functions. III

Summary This paper is a continuation of [2, 3]. It contains anALGOL program for the incomplete elliptic integral of the third kind based on a theory described in [4]. This program replaces the inadequate one based on the Gauß-transformation which was published in [2]. In addition, anAlgol program for a general complete elliptic integral is presented. Editor's note. In this fascicle, prepublication of algorithms from the Special Functions Series of the Handbook for Automatic Computation is continued. Algorithms are published inAlgol 60 reference language as approved by the IFIP. Contributions in this series should be styled after the most recently published ones.This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the U. S. Army Research Office —Durham under Contract DA-31-124-ARO-D-257.     

Inequalities for the generalized elliptic integrals and modular functions

In this paper, the authors study monotonicity and convexity of the generalized elliptic integrals and certain combinations of these special functions, such as ma(r) and μa(r). Making use of these results, the authors obtain some sharp inequalities for the so-called Ramanujan's generalized modular functions.     

Legendre functions,spherical rotations,and multiple elliptic integrals

A closed-form formula is derived for the generalized Clebsch–Gordan integral \(\int_{-1}^{1} {[}P_{\nu}(x){]}^{2}P_{\nu}(-x)\,\mathrm {d}x\) , with P _ν being the Legendre function of arbitrary complex degree \(\nu\in\mathbb{C}\) . The finite Hilbert transform of P _ν (x)P _ν (?x), ?1<x<1 is evaluated. An analytic proof is provided for a recently conjectured identity \(\int_{0}^{1}[\mathbf{K}( \sqrt{1-k^{2}} )]^{3}\,\mathrm {d}k=6\int_{0}^{1}[\mathbf{K}(k)]^{2}\mathbf{K}( \sqrt{1-k^{2}} )k\,\mathrm {d}k=[\Gamma (\frac{1}{4})]^{8}/(128\pi^{2}) \) involving complete elliptic integrals of the first kind K(k) and Euler’s gamma function Γ(z).     

Algorithms for computing complete elliptic integrals and some related functions

We propose some new algorithms for computing the complete elliptic integrals of the first and second kinds and some related functions. The algorithms are constructed from rapidly converging power series; the sign-definiteness of the terms of the series guarantees their good conditionality (stability with respect to rounding errors). The algorithms turned out flexible and easily adjustable to every specific demand of computational mathematics.     

Non-oscillation of elliptic integrals

Moscow Radioelectronics and Automatics Institute. Translated from Funktsyonal'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 3, pp. 45–50, July–September, 1990.     

Special polynomials and elliptic integrals

We show that the use of generalized multivariable forms of Hermite polynomials provide a useful tool for the evaluation of families of elliptic type integrals often encountered in electrostatics and electrodynamics     

The commutativity of integrals of motion for quantum spin chains and elliptic functions identities

We prove the commutativity of the first two nontrivial integrals of motion for quantum spin chains with elliptic form of the exchange interaction. We also show their liner independence for the number of spins larger than 4. As a byproduct, we obtained several identities between elliptic Weierstrass functions of three and four arguments.      

Determinants of elliptic hypergeometric integrals

We start from an interpretation of the BC ₂-symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation and then generalize this construction to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding formulas for the elliptic beta integral and symmetry transformation in a new way, by proving that both sides satisfy the same difference equations and that these difference equations satisfy a needed Galois-theoretic condition ensuring the uniqueness of the simultaneous solution.     

Limits of elliptic hypergeometric integrals

In Ann. Math., to appear, 2008, the author proved a number of multivariate elliptic hypergeometric integrals. The purpose of the present note is to explore more carefully the various limiting cases (hyperbolic, trigonometric, rational, and classical) that exist. In particular, we show (using some new estimates of generalized gamma functions) that the hyperbolic integrals (previously treated as purely formal limits) are indeed limiting cases. We also obtain a number of new trigonometric (q-hypergeometric) integral identities as limits from the elliptic level. The author was supported in part by NSF Grant No. DMS-0401387.     

Inequalities and bounds for elliptic integrals

Computable lower and upper bounds for the symmetric elliptic integrals and for Legendre's incomplete integral of the first kind are obtained. New bounds are sharper than those known earlier. Several inequalities involving integrals under discussion are derived.     

Bounds for symmetric elliptic integrals

Lower and upper bounds for the four standard incomplete symmetric elliptic integrals are obtained. The bounding functions are expressed in terms of the elementary transcendental functions. Sharp bounds for the ratio of the complete elliptic integrals of the second kind and the first kind are also derived. These results can be used to obtain bounds for the product of these integrals. It is shown that an iterative numerical algorithm for computing the ratios and products of complete integrals has the second order of convergence.     

A certain class of incomplete elliptic integrals and associated definite integrals

In many seemingly diverse physical contexts (including, for example, certain radiation field problems, studies of crystallographic minimal surfaces, the theory of scattering of acoustic or electromagnetic waves by means of an elliptic disk, studies of elliptical crack problems in fracture mechanics, and so on), a remarkably large number of general families of elliptic-type integrals, and indeed also many definite integrals of such families with respect to their modulus (or complementary modulus), are known to arise naturally. Motivated essentially by these and many other potential avenues of their applications, we present here a systematic account of the theory of a certain family of incomplete elliptic integrals in a unified and generalized manner. By means of the familiar Riemann–Liouville fractional differintegral operators, we obtain several explicit hypergeometric representations and apply these representations with a view to deriving various associated definite integrals, not only with respect to the modulus (or complementary modulus), but also with respect to the amplitude of the incomplete elliptic integrals involved therein.     

Integrals involving complete elliptic integrals

We give a closed-form evaluation of a number of Erdélyi-Kober fractional integrals involving elliptic integrals of the first and second kind, in terms of the ₃F₂ generalized hypergeometric function. Reduction formulae for ₃F₂ enable us to simplify the solutions for a number of particular cases.     

Computing elliptic integrals by duplication

Summary Logarithms, arctangents, and elliptic integrals of all three kinds (including complete integrals) are evaluated numerically by successive applications of the duplication theorem. When the convergence is improved by including a fixed number of terms of Taylor's series, the error ultimately decreases by a factor of 4096 in each cycle of iteration. Except for Cauchy principal values there is no separation of cases according to the values of the variables, and no serious cancellations occur if the variables are real and nonnegative. Only rational operations and square roots are required. An appendix contains a recurrence relation and two new representations (in terms of elementary symmetric functions and power sums) forR-polynomials, as well as an upper bound for the error made in truncating the Taylor series of anR-function.