1  Introduction

(1) The integrals were first converted, by various substitutions that differ from one integrand to another, to integrals of Jacobian elliptic functions, which were then expressed separately in terms of Legendre's canonical forms. Two Russian tables [ Gradshteyn and Ryzhik1994, Prudnikov et al. 1986] later omitted the intermediate integrals of Jacobian elliptic functions and did not specify methods of integration.

(2) Each interval of integration began or ended at a branch point of the integrand. An integral with neither limit of integration at a branch point would have to be split into two parts, doubling the number of canonical forms, and even this remedy was not always available because of divergence at both neighboring branch points.

(3) The ordering of the free limit of integration and the branch points on the real line had to be specified. In conjunction with (2) this often required listing 8, 18, 36, or even 72 cases of what was essentially the same integral. The formulas for these numerous cases often had little resemblance to one another, differing by an integral of the first kind or an algebraic function or both, and gave no hint of how they might be unified. The problem is most easily seen in the arrangement chosen by .

investigated four possible approaches to symbolic integration of elliptic integrals and found serious difficulties in each case. Besides the method of Byrd and Friedman they considered direct conversion to the canonical forms of Legendre and of Weierstrass. In the fourth method, elaborated earlier by , an elliptic integral was represented as a multivariate hypergeometric R-function before being reduced to canonical forms by recurrence relations. Although the R-functions offered certain advantages, a major difficulty lay in multiparameter recurrence relations. settled on a first stage of reduction by a classical method using one-parameter recurrence relations, to be followed by a second stage of expression in terms of R-functions or alternatively Legendre's integrals. Implementation was not free of troubles.

In a paper not yet published, choose Legendre's canonical forms but make improvements on the classical method. They use Hermite reduction [ Moses1971][ Geddes et al. 1992] to arrive at integrands without multiple poles, and the terms with simple poles are decomposed by an implicit full partial-fraction expansion due to . To reduce to Legendre's canonical forms they finally choose one of 21 transformations according to the zeros of s² and the limits of integration.

In the present paper we separate two stages of reduction, as did . In the first stage we start from an integrand symbolically factored into possibly complex linear factors, permitting explicit formulas for reduction by one-parameter recurrence relations to a set of basic integrals somewhat different from the classical ones. All coefficients can remain symbolic throughout this reduction, which is relatively simple for integrands that occur frequently in practice. For integrands containing polynomials of high degree, like some of the examples chosen by Labahn and Mutrie, the iteration of recurrence relations could become onerous, and it might be better to begin, as they do, with a preliminary Hermite reduction before applying the present method to the resulting individual terms.

In the second stage the basic integrals are expressed in terms of canonical R-functions, using reduction theorems unknown at the time of Ng and Polajnar's work. These make it possible to avoid the complications in (2) above and unify the numerous cases mentioned in (3) (e.g. see [ Carlson1988]) while still retaining symbolic coefficients, some of which may be complex. Although the 21 transformations used by are avoided, and although there are efficient algorithms [ Carlson1995] for computing numerically the canonical R-functions of complex variables, the possible presence of complex numbers will seem a disadvantage to those who use a computer language without complex arithmetic and to those who demand that a real integrand have an integral that contains no complex numbers. The complex numbers can be eliminated by a Landen transformation (e.g. see [ Carlson1991]), but only at the cost of more complicated formulas. On the other hand an integrand containing only linear factors with symbolic coefficients can be integrated in terms of canonical R-functions with no assumptions about the numerical values of the symbols; for this purpose Legendre's canonical forms are unsuitable.

A simple rationalization procedure, found in every introduction to the subject, allows us to separate from (1.1) the integral of a rational function and assume that r(s,t) = r(t)/s, where r is a rational function of t. (Moreover, if s² is an even function of t, it is advantageous but not essential to separate the odd part of r, which leads to an elementary integral, and replace t by Öt in the remaining elliptic integral.) Instead of (1.1) we henceforth consider

ó õ x y   P(t) dt Q(t)s(t) = ó õ x y  h Õ i=1  (ai +bi t)-1/2 n Õ j=1  (aj +bj t)mj dt ,

(2)

where P and Q are polynomials, h=3 or 4, and the m's are integers. We assume that the integral is well defined (in particular that the open interval of integration contains no finite branch point of the integrand) and that no two of the n linear factors on the right side are proportional. Although the a's and b's, some of which may be complex, can be left as symbols throughout the integration procedure described in this paper, the integer values of the m's are needed at the beginning. In many cases, including most of those in present integral tables, the integrand contains contains only linear and quadratic factors, making the m's obvious. If unfactored polynomials of higher degree are present, we must not only ensure that P and Q have no common divisor (except a constant) but also determine whether they have divisors in common with s². If so, and if all coefficients are given numerically, the m's can be determined by making square-free factorizations (, § 8.2) of P and Q and finding the greatest common divisor of each square-free factor and s², which is square-free by assumption.