A recent table of elliptic integrals [9,10,11,12,13] uses symmetric standard integrals instead of Legendre's integrals because permutation symmetry makes it possible to unify many of the formulas in previous tables. Fortran codes for numerical computation of the symmetric integrals, which are homogeneous functions of three or four variables, can be found in several major software libraries as well as in the supplements to [9,10]. For analytical purposes it is desirable to know how the homogeneous functions behave when some of the variables are much larger than the others. For all such cases we list in Section 2 asymptotic approximations (sometimes two or three approximations of different accuracy), always with error bounds. Proofs are discussed in Section 3. In most cases the approximations are obtained by replacing the integrand by a uniform approximation. Many of the results found by a different method in [16] have been improved by sharpening the error bounds or by finding bounds for incomplete elliptic integrals that are still useful for the complete integrals, which are then not listed separately. Cases not considered in [16] include two for a completely symmetric integral of the second kind and two for a symmetric integral of the third kind in which two variables are much larger than the other two.
We assume that x,y,z are nonnegative and at most one of them is 0. The symmetric integral of the first kind,
is homogeneous of degree -1/2 in x,y,z and satisfies RF(x,x,x) = x-1/2. The symmetric integral of the third kind,
is homogeneous of degree -3/2 in x,y,z,p and satisfies RJ(x,x,x,x) = x-3/2. If p=z, RJ reduces to an integral of the second kind,
which is symmetric in x and y only. If two variables of RF are equal, the integral becomes an elementary function,
If x< y it is an inverse trigonometric function,
and if x> y it is an inverse hyperbolic function,
If the second argument of RC is negative, the Cauchy principal value is [18, (4.8),]
If the fourth argument of RJ is negative, the Cauchy principal value is given by [18, (4.6),]
where q-y = (z-y) (y-x) / (y+p). If we permute the values of x,y,z so
that (z-y) (y-x) 
0, then (q y > 0 .
A completely symmetric integral of the second kind is not as convenient as RD for use in tables because its representation by a single integral is more complicated [7, (9.1-9),]:
It is symmetric and homogeneous of degree 1/2 in x, y, z, and it satisfies RG(x, x, x) = x1/2 . It has a nice representation by a double integral that expresses the surface area of an ellipsoid [7, (9.4-6),]. It is related to RD and RF by (58) and by
Legendre's complete elliptic integrals K and E are given by
Approximations and inequalities for K, E, and some combinations thereof are given in [1,2,3]. If the error terms in (30), ,(31), and (53) are omitted, the approximations reduce to the leading terms of well-known series expansions of K and E for k near 1 [15, p. 54,] [4, 900.06, 900.10,]. If the series for K is truncated after any number of terms, simple bounds for the relative error are given in [14, (1.17),]. A generalization of this series to RF(x, y, z) with x, y < < z is given in [14, (1.14)-(1.16),], again with simple bounds for the relative error of truncation.
The various functions designated by R with a letter subscript are special cases of the multivariate hypergeometric R-function,
which is symmetric in the indices 1,...,n and homogeneous of degree -a in the variables z1, ..., zn. Best regarded as the Dirichlet average of x-a [7, § 5.9,], it is a symmetric variant of the function known as Lauricella's FD. By the method of Mellin transforms, series expansions are obtained in [8, (4.16)-(4.19),] that converge rapidly if some of the z's are much larger than the others and if the parameters satisfy i=1nbi > a > 0 . Thus the leading terms of these series provide asymptotic approximations for all except RG among the functions
However, error bounds for the approximations are more easily derived by the methods of the present paper. Another function that is used repeatedly in obtaining error bounds is [7, Ex. 9.8-5,]
In Section 4 the asymptotic approximations are applied to show that RF(x, y, z) , RD(x, y, z), RJ(x, y, z, p), and xyz-1/2 are linearly independent with respect to coefficients that are rational functions of x,y,z, and p. An Appendix contains some elementary inequalities that are used in obtaining error bounds.
The results in this paper provide upper and lower approximations that approach the elliptic integrals as selected ratios of the variables approach zero. Approximations that approach the integrals as all variables approach a common value have been found by other methods. For example, the theory of hypergeometric mean values yields upper and lower algebraic approximations for all the integrals in this paper [5, Thm. 2,], while truncation of Taylor series about the arithmetic mean of the variables gives approximations with errors that may be positive or negative. Successive applications of the duplication theorem for RF, making its three variables approach equality, provide ascending and descending sequences of successively sharper (and successively more complicated) algebraic approximations to RF and RC [6]. Transcendental approximations that approach RF when only two of its variables approach equality are furnished by
which follows from (71). The inequalities can be sharpened by first using Landen or Gauss transformations of RF [7, § 9.5,] to make y and z approach equality. If x = 0 the Gauss transformation reduces to replacing 
and 
by their arithmetic and geometric means, and each RC-function becomes 
/2 divided by the square root of its second argument. Therefore, in the complete case the procedure reduces to the algorithm of the arithmetic-geometric mean [7, (6.10-6)(9.2-3),] and provides ascending and descending sequences of algebraic approximations, of which leading members are shown in (33).
