Calculation of the complete elliptic integrals with complex modulus

Summary This is an Addendum to a preceding paper of Morita and Horiguchi [Numer. Math.20, 425–430 (1973)]. Attention is called to an error in the algol procedure given in that paper. A corrected procedure of calculating the complete elliptic integrals of the first and the second kind with complex modulusk is presented, in the form that is itself useful in the calculation of their analytic continuations over the branch cuts.     

Calculation of the elliptic integrals of the first and second kinds with complex modulus

Summary. Numerical procedures for calculating the elliptic integrals of the first and the second kind with complex modulus and their analytic continuations are presented. The corresponding results for the elliptic integral of the third kind are given in Appendices. Received September 9, 1996 / Revised version received June 2, 1998     

Some functional inequalities for complete elliptic integrals

We establish some monotonicity results and inequalities involving elliptic integrals of the first and the second kind. Work sponsored by the Ministero dell’Università e della Ricerca Scientifica e Tecnologica of Italy.     

Inequalities and bounds for generalized complete elliptic integrals

Computable bounds for the generalized complete elliptic integrals of the first and second kind are obtained. Also, bounds for some combinations and products for integrals under discussion are established. It has been proven that both families of integrals are logarithmically convex as functions of the first parameter. This property has been employed to obtain several inequalities involving integrals in question.     

Bounds for complete elliptic integrals of the first kind

In this note by using some elementary computations we present some new sharp lower and upper bounds for the complete elliptic integrals of the first kind. These results improve some known bounds in the literature and are deduced from the well-known Wallis inequality, which has been studied extensively in the last 10 years.     

Indefinite integrals involving complete elliptic integrals of the third kind

A method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete elliptic integrals. This gives a total of 27 integrals for the complete integral of the third kind, including the single integral given in the literature. Some typographical errors in a previous related paper are corrected.     

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.     

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.     

Sharp power-type Heronian and Lehmer means inequalities for the complete elliptic integrals

In the article, we prove that the inequalities     

Hypergeometric analogues of the arithmetic-geometric mean iteration

The arithmetic-geometric mean iteration of Gauss and Legendre is the two-term iterationa _n +1=(a _n +b _n )/2 and \(b_{n + 1} = \sqrt {a_n b_n } \) witha ₀?1 andb ₀?x. The common limit is₂ F ₁(1/2, 1/2; 1; 1?x ²)^?1 and the convergence is quadratic. This is a rare object with very few close relatives. There are however three other hypergeometric functions for which we expect similar iterations to exist, namely:₂ F ₁(1/2?s ₁, 1/2+s; 1; ·) withs=1/3, 1/4, 1/6. Our intention is to exhibit explicitly these iterations and some of their generalizations. These iterations exist because of underlying quadratic or cubic transformations of certain hypergeometric functions, and thus the problem may be approached via searching for invariances of the corresponding second-order differential equations. It may also be approached by searching for various quadratic and cubic modular equations for the modular forms that arise on inverting the ratios of the solutions of these differential equations. In either case, the problem is intrinsically computational. Indeed, the discovery of the identities and their proofs can be effected almost entirely computationally with the aid of a symbolic manipulation package, and we intend to emphasize this computational approach.     

Turán type inequalities for generalized complete elliptic integrals

In this paper our aim is to establish some Turán type inequalities for Gaussian hypergeometric functions and for generalized complete elliptic integrals. These results complete the earlier result of P. Turán proved for Legendre polynomials. Moreover we show that there is a close connection between a Turán type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind. At the end of this paper we prove a recent conjecture of T. Sugawa and M. Vuorinen related to estimates of the hyperbolic distance of the twice punctured plane. Dedicated to my son Koppány.     

Convexity properties of generalizations of the arithmetic-geometric mean

In the eighteenth century, Landen, Lagrange and Gauss studied a function of two positive real numbers that has become known as the arithmetic-geometric mean (AGM). In the nineteenth century, Borchardt generalized the AGM to a function of any 2ⁿ(n = 1,2,3,…) positive real numbers. In this paper, we generalize the AGM to a function of any even number of positive real numbers. If M(a, b) is the original AGM then M(a, b) is concave in the pair (a, b) of positive numbers and log M(e^α, e^β) is convex in the pair (α,β) of real numbers; all our generalizations of the AGM behave similarly. We generalize this analysis extensively.     

Numerical computation of real or complex elliptic integrals

Algorithms for numerical computation of symmetric elliptic integrals of all three kinds are improved in several ways and extended to complex values of the variables (with some restrictions in the case of the integral of the third kind). Numerical check values, consistency checks, and relations to Legendre's integrals and Bulirsch's integrals are included.This work was supported by the Director of Energy Research, Office of Basic Energy Sciences. The Ames Laboratory is operated for the US Department of Energy by Iowa State University under Contract W-7405-ENG-82.     

A relationship between subpermanents and the arithmetic-geometric mean inequality

Using the arithmetic-geometric mean inequality, we give bounds for k-subpermanents of nonnegative n×n matrices F. In the case k=n, we exhibit an n²-set S whose arithmetic and geometric means constitute upper and lower bounds for per(F)/n!. We offer sharpened versions of these bounds when F has zero-valued entries.