For relations of Legendre’s integrals to theta functions, Jacobian functions,
and Weierstrass functions, see §§20.9(i), 22.15(ii),
and 23.6(iv), respectively. See also
Erdélyi et al. (1953b, Chapter 13).
If is assumed (without loss of generality), then
In each case when , the quantity multiplying supplies the
asymptotic behavior of the left-hand side as the left-hand side tends to 0.
For relations to the Gudermannian function and its
inverse (§4.23(viii)), see (19.6.8)