# §19.16 Definitions

## §19.16(i) Symmetric Integrals

19.16.1
19.16.2
19.16.3

where () is a real or complex constant, and

19.16.4

In (19.16.1) and (19.16.2), except that one or more of may be 0 when the corresponding integral converges. In (19.16.2) the Cauchy principal value is taken when is real and negative. In (19.16.3) , , . It should be noted that the integrals (19.16.1)–(19.16.3) have been normalized so that .

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

19.16.5

with the same conditions on , , as for (19.16.1), but now .

Just as the elementary function 19.2(iv)) is the degenerate case

and is a degenerate case of , so is a degenerate case of the hyperelliptic integral,

19.16.7

## §19.16(ii)

All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.16.3), are special cases of a multivariate hypergeometric function

which is homogeneous and of degree in the ’s, and unchanged when the same permutation is applied to both sets of subscripts . Thus is symmetric in the variables and if the parameters and are equal. The -function is often used to make a unified statement of a property of several elliptic integrals. Before 1969 was denoted by .

where is the beta function (§5.12) and

19.16.10
19.16.11
,.

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

## §19.16(iii) Various Cases of

is an elliptic integral iff the ’s are distinct and exactly four of the parameters are half-odd-integers, the rest are integers, and none of , , is zero or a negative integer. The only cases that are integrals of the first kind are the four in which each of and is either or 1 and each is . The only cases that are integrals of the third kind are those in which at least one is a positive integer. All other elliptic cases are integrals of the second kind.

(Note that is not an elliptic integral.)

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