§19.16 Definitions

where $p$ ($\ne 0$) is a real or complex constant, and

In (19.16.1)–(19.16.2_5), $x,y,z\in ℂ\setminus (-\mathrm{\infty },0]$ except that one or more of $x,y,z$ may be 0 when the corresponding integral converges. In (19.16.2) the Cauchy principal value is taken when $p$ is real and negative. See also (19.20.14). It should be noted that the integrals (19.16.1)–(19.16.2_5) have been normalized so that $R_F(1,1,1)=R_J(1,1,1,1)=R_G(1,1,1)=1$.

A fourth integral that is symmetric in only two variables is defined by

with the same conditions on $x$, $y$, $z$ as for (19.16.1), but now $z\ne 0$.

Just as the elementary function $R_C(x,y)$ (§19.2(iv)) is the degenerate case

and $R_D$ is a degenerate case of $R_J$, so is $R_J$ a degenerate case of the hyperelliptic integral,

All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function

which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\mathrm{\dots },n$. Thus $R_{-a}(𝐛;𝐳)$ is symmetric in the variables $z_j$ and $z_{\mathrm{\ell }}$ if the parameters $b_j$ and $b_{\mathrm{\ell }}$ are equal. The $R$-function is often used to make a unified statement of a property of several elliptic integrals. Before 1969 $R_{-a}(𝐛;𝐳)$ was denoted by $R(a;𝐛;𝐳)$.

where $\mathrm{B}(x,y)$ is the beta function (§5.12) and

When $n=4$ a useful version of (19.16.9) is given by

where

For generalizations and further information, especially representation of the $R$-function as a Dirichlet average, see Carlson (1977b).

$R_{-a}(𝐛;𝐳)$ is an elliptic integral iff the $z$’s are distinct and exactly four of the parameters $a,a^{\prime },b_1,\mathrm{\dots },b_n$ are half-odd-integers, the rest are integers, and none of $a$, $a^{\prime }$, $a+a^{\prime }$ is zero or a negative integer. The only cases that are integrals of the first kind are the four in which each of $a$ and $a^{\prime }$ is either $\frac{1}{2}$ or 1 and each $b_j$ is $\frac{1}{2}$. The only cases that are integrals of the third kind are those in which at least one $b_j$ is a positive integer. All other elliptic cases are integrals of the second kind.

(Note that $R_C(x,y)$ is not an elliptic integral.)

When one variable is 0 without destroying convergence, any one of (19.16.14)–(19.16.17) is said to be complete and can be written as an $R$-function with one less variable:

Thus

The last $R$-function has $a=a^{\prime }=\frac{1}{2}$.

Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral: