Given real or complex numbers , with not real and negative, define
for , where the square root is chosen so that , where and are chosen so that their difference is numerically less than . Then as sequences , converge to a common limit , the arithmetic-geometric mean of . And since
convergence is very rapid.
For real and , use (22.20.1) with , , , and continue until is zero to the required accuracy. Next, compute , where
and the inverse sine has its principal value (§4.23(ii)). Then
and the subsidiary functions can be found using (22.2.10).
See also Wachspress (2000).
By application of the transformations given in §§22.7(i) and 22.7(ii), or can always be made sufficently small to enable the approximations given in §22.10(ii) to be applied. The rate of convergence is similar to that for the arithmetic-geometric mean.
If either or is given, then we use , , , and , obtaining the values of the theta functions as in §20.14.
If are given with and , then can be found from
using the arithmetic-geometric mean.
If , then four iterations of (22.20.1) give .
See Wachspress (2000).