text: Elliptic Integrals are said to be 'complete' when the amplitude  and therefore . The complete elliptic integral of the first kind  may thus be defined as
  or more compactly in terms of the incomplete integral of the first kind as
  It can be expressed as a power series
  where  is the Legendre polynomials, which is equivalent to
  where  denotes the double factorial. In terms of the Gauss hypergeometric function, the complete elliptic integral of the first kind can be expressed as
  The complete elliptic integral of the first kind is sometimes called the quarter period. It can be computed very efficiently in terms of the arithmetic–geometric mean:
  Therefore the modulus can be transformed as:
  This expression is valid for all  and :
  Relation to the gamma function
  If  and  (where  is the modular lambda function), then  is expressible in closed form in terms of the gamma function. For example, ,  and  give, respectively,
  and
  and
  More generally, the condition that
  be in an imaginary quadratic field is sufficient. For instance, if , then  and
  Relation to Jacobi theta function
  The relation to Jacobi's theta function is given by
  where the nome  is
  Asymptotic expressions
  This approximation has a relative precision better than  for . Keeping only the first two terms is correct to 0.01 precision for .