## §19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)

When and are positive numbers, define

19.8.1
,.

As , and converge to a common limit called the AGM (Arithmetic-Geometric Mean) of and . By symmetry in and we may assume and define

19.8.2

Then

19.8.3

showing that the convergence of to 0 and of and to is quadratic in each case.

The AGM appears in

19.8.6, , ,

and in

where , , , , and

19.8.8
,.

Again, and converge quadratically to and 0, respectively, and converges to 0 faster than quadratically. If , then the Cauchy principal value is

where (19.8.8) still applies, but with

## §19.8(iii) Gauss Transformation

We consider only the descending Gauss transformation because its (ascending) inverse moves closer to the singularity at . Let

(Note that and imply and , and also that implies , thus preserving completeness.) Then

19.8.19

where

If , then is pure imaginary.