Let the Modulus satisfy . (This may also be written in terms of
the Parameter or Modular Angle
.) The incomplete elliptic integral
of the second kind is then defined as

(1) |

(2) |

(3) | |||

(4) |

so the elliptic integral can also be written as

(5) |

The complete elliptic integral of the second kind, illustrated above as a function of the Parameter ,
is defined by

(6) | |||

(7) | |||

(8) | |||

(9) |

where is the Hypergeometric Function and is a Jacobi Elliptic Function. The complete elliptic integral of the second kind satisfies the Legendre Relation

(10) |

(11) |

(12) |

(13) |

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Elliptic Integrals.'' Ch. 17 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 587-607, 1972.

Spanier, J. and Oldham, K. B. ``The Complete Elliptic Integrals and '' and
``The Incomplete Elliptic Integrals and .''
Chs. 61 and 62 in *An Atlas of Functions.* Washington, DC: Hemisphere, pp. 609-633, 1987.

Whittaker, E. T. and Watson, G. N. *A Course in Modern Analysis, 4th ed.* Cambridge, England:
Cambridge University Press, 1990.

© 1996-9

1999-05-25