## §19.22(i) Complete Integrals

### ¶ Bartky’s Transformation

If , then (19.22.7) reduces to (19.22.3), but if or , then both sides of (19.22.4) are 0 by (19.20.9). If or , then and are complex conjugates.

## §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

19.22.11

has the same sign as for .

where and

19.22.13

(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

19.22.15

## §19.22(iii) Incomplete Integrals

Let , , and have positive real parts, assume , and retain (19.22.5) and (19.22.6). Define

19.22.16

so that

19.22.17

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

19.22.23

and the corresponding equations with , , and replaced by , , and , respectively. These relations need to be used with caution because is negative when .