Let , , , and . Then
has the same sign as for .
(If , then and (19.22.13) reduces to (19.22.11).) As , and converge quadratically to and 0, respectively, and converges to 0 faster than quadratically. If the last variable of is negative, then the Cauchy principal value is
and (19.22.13) still applies, provided that
If are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when (implying ), and descending Gauss transformations when (implying ). Ascent and descent correspond respectively to increase and decrease of in Legendre’s notation. Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not.
If or , then (19.22.20) reduces to by (19.20.13), and if or then (19.22.19) reduces to by (19.20.20) and (19.22.22). If or , then and are complex conjugates. However, if and are complex conjugates and and are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and .
The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. The equations inverse to (19.22.5) and (19.22.16) are given by
and the corresponding equations with , , and replaced by , , and , respectively. These relations need to be used with caution because is negative when .