text: With  one has: demonstrating that this Jacobian elliptic function is a simple inverse of the incomplete elliptic integral of the first kind.
  The incomplete elliptic integral of the first kind has following addition theorem:
  The elliptic modulus can be transformed that way:
  Incomplete elliptic integral of the second kind
  The incomplete elliptic integral of the second kind  in trigonometric form is
  Substituting  and , one obtains the Legendre normal form:
  Equivalently, in terms of the amplitude and modular angle:
  Relations with the Jacobi elliptic functions include
  The meridian arc length from the equator to latitude  is written in terms of :
  where  is the semi-major axis, and  is the eccentricity.
  The incomplete elliptic integral of the second kind has following addition theorem:
  The elliptic modulus can be transformed that way:
  Incomplete elliptic integral of the third kind
  The incomplete elliptic integral of the third kind  is
  or
  The number  is called the characteristic and can take on any value, independently of the other arguments. Note though that the value  is infinite, for any .
  A relation with the Jacobian elliptic functions is
  The meridian arc length from the equator to latitude  is also related to a special case of :
  Complete elliptic integral of the first kind