text: Differential equation The differential equation for the elliptic integral of the first kind is
  A second solution to this equation is . This solution satisfies the relation
  Continued fraction
  A continued fraction expansion is:
  where the nome is  in its definition.
  Complete elliptic integral of the second kind
  The complete elliptic integral of the second kind  is defined as
  or more compactly in terms of the incomplete integral of the second kind  as
  For an ellipse with semi-major axis  and semi-minor axis  and eccentricity , the complete elliptic integral of the second kind  is equal to one quarter of the circumference  of the ellipse measured in units of the semi-major axis . In other words:
  The complete elliptic integral of the second kind can be expressed as a power series
  which is equivalent to
  In terms of the Gauss hypergeometric function, the complete elliptic integral of the second kind can be expressed as
  The modulus can be transformed that way:
  Computation
  Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the arithmetic–geometric mean.
  Define sequences  and , where ,  and the recurrence relations ,  hold. Furthermore, define
  By definition,
  Also