where () is a real or complex constant, and
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
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,
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
When a useful version of (19.16.9) is given by
For further information, especially representation of the -function as a Dirichlet average, see Carlson (1977b).
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.)
The last -function has .