When and are positive numbers, define
As , and converge to a common limit called the AGM (Arithmetic-Geometric Mean) of and . By symmetry in and we may assume and define
showing that the convergence of to 0 and of and to is quadratic in each case.
The AGM has the integral representations
The first of these shows that
The AGM appears in
where , , , , and
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
(Note that and imply and , and also that implies .) Then
(Note that and imply and .) Then
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
If , then is pure imaginary.