The approximations in §§19.27(i)–19.27(v) are
furnished with upper and lower bounds by Carlson and Gustafson (1994),
sometimes with two or three approximations of differing accuracies. Although
they are obtained (with some exceptions) by approximating uniformly the
integrand of each elliptic integral, some occur also as the leading terms of
known asymptotic series with error bounds (Wong (1983, §4),
Carlson and Gustafson (1985), López (2000, 2001)). These series
converge but not fast enough, given the complicated nature of their terms, to
be very useful in practice.
A similar (but more general) situation prevails for
when some of the variables
are smaller in magnitude than the rest;
see Carlson (1985, (4.16)–(4.19) and (2.26)–(2.29)).