text: Then In practice, the arithmetic-geometric mean would simply be computed up to some limit. This formula converges quadratically for all . To speed up computation further, the relation  can be used.
  Furthermore, if  and  (where  is the modular lambda function), then  is expressible in closed form in terms of
  and hence can be computed without the need for the infinite summation term. For example, ,  and  give, respectively,
  and
  and
  Derivative and differential equation
  A second solution to this equation is .
  Complete elliptic integral of the third kind
  The complete elliptic integral of the third kind  can be defined as
  Note that sometimes the elliptic integral of the third kind is defined with an inverse sign for the characteristic ,
  Just like the complete elliptic integrals of the first and second kind, the complete elliptic integral of the third kind can be computed very efficiently using the arithmetic-geometric mean.
  Partial derivatives
  Jacobi zeta function
  In 1829, Jacobi defined the Jacobi zeta function:
  It is periodic in  with minimal period . It is related to the Jacobi zn function by . In the literature (e.g. Whittaker and Watson (1927)), sometimes  means Wikipedia's . Some authors (e.g. King (1924)) use  for both Wikipedia's  and .