19.16.1 | |||
19.16.2 | |||
19.16.2_5 | |||
19.16.3 | Moved to (19.23.6_5). | ||
where () is a real or complex constant, and
19.16.4 | |||
In (19.16.1)–(19.16.2_5), 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. See also (19.20.14). It should be noted that the integrals (19.16.1)–(19.16.2_5) 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
19.16.6 | |||
and is a degenerate case of , so is a degenerate case of the hyperelliptic integral,
19.16.7 | |||
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
19.16.8 | |||
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 .
19.16.9 | |||
, , , | |||
where is the beta function (§5.12) and
19.16.10 | |||
19.16.11 | ||||
, | ||||
. | ||||
For generalizations and 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.
19.16.14 | ||||
19.16.15 | ||||
19.16.16 | ||||
19.16.17 | ||||
19.16.18 | ||||
(Note that is not an elliptic integral.)