Let $\mathrm{\Re }x>0$, $\mathrm{\Re }y>0$, $a=(x+y)/2$, and $p\ne 0$. Then

where

and hence

The AGM, $M(a_0,g_0)$, of two positive numbers $a_0$ and $g_0$ is defined in §19.8(i). Again, we assume that $a_0\ge g_0$ (except in (19.22.10)), and define $c_n=\sqrt{a_n^2-g_n^2}$. Then

and

where

$Q_n$ has the same sign as $a_0-g_0$ for $n\ge 1$.

where $p_0>0$ and

(If $p_0=a_0$, then $p_n=a_n$ and (19.22.13) reduces to (19.22.11).) As $n\to \mathrm{\infty }$, $p_n$ and ${\epsilon }_n$ converge quadratically to $M(a_0,g_0)$ and 0, respectively, and $Q_n$ converges to 0 faster than quadratically. If the last variable of $R_J$ is negative, then the Cauchy principal value is

and (19.22.13) still applies, provided that

Let $x$, $y$, and $z$ have positive real parts, assume $p\ne 0$, and retain (19.22.5) and (19.22.6). Define

so that

Then

If $x,y,z$ 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 $k$ in Legendre’s notation. Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not.

If $p=x$ or $p=y$, then (19.22.20) reduces to $0=0$ by (19.20.13), and if $z=x$ or $z=y$ then (19.22.19) reduces to $0=0$ by (19.20.20) and (19.22.22). If or , then $z_{+}$ and $z_{-}$ are complex conjugates. However, if $x$ and $y$ are complex conjugates and $z$ and $p$ 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 $p_{\pm }^2-p_{\mp }^2=\pm |p^2-x^2|\ne 0$.

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 $z$, $z_{+}$, and $z_{-}$ replaced by $p$, $p_{+}$, and $p_{-}$, respectively. These relations need to be used with caution because $y$ is negative when .