§19.22 Quadratic Transformations
Contents
- §19.22(i) Complete Integrals
- §19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)
- §19.22(iii) Incomplete Integrals
§19.22(i) Complete Integrals
Let
,
,
, and
. Then
where
and hence
§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)
The AGM,
, of two positive numbers
and
is defined
in §19.8(i). Again, we assume that
(except in
(19.22.10)), and define
. Then
and
where
has the same sign as
for
.
where
and
(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
§19.22(iii) Incomplete Integrals
Let
,
, and
have positive real parts, assume
, and retain
(19.22.5) and (19.22.6). Define
so that
Then
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
.

