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 y>0 is assumed (without loss of generality), then
In each case when y=1, the quantity multiplying RC supplies the
asymptotic behavior of the left-hand side as the left-hand side tends to 0.
For relations to the Gudermannian function gd(x) and its
inverse gd-1(x) (§4.23(viii)), see (19.6.8)