text: There are still other conventions for the notation of elliptic integrals employed in the literature. The notation with interchanged arguments, , is often encountered; and similarly  for the integral of the second kind. Abramowitz and Stegun substitute the integral of the first kind, , for the argument  in their definition of the integrals of the second and third kinds, unless this argument is followed by a vertical bar: i.e.  for . Moreover, their complete integrals employ the parameter  as argument in place of the modulus , i.e.  rather than . And the integral of the third kind defined by Gradshteyn and Ryzhik, , puts the amplitude  first and not the "characteristic" .
  Thus one must be careful with the notation when using these functions, because various reputable references and software packages use different conventions in the definitions of the elliptic functions. For example, Wolfram's Mathematica software and Wolfram Alpha define the complete elliptic integral of the first kind in terms of the parameter , instead of the elliptic modulus .
  Incomplete elliptic integral of the first kind
  The incomplete elliptic integral of the first kind  is defined as
  This is the trigonometric form of the integral; substituting  and , one obtains the Legendre normal form:
  Equivalently, in terms of the amplitude and modular angle one has: