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

(1) |

(2) | |||

(3) |

(4) |

Let

(5) | |||

(6) |

so the integral can also be written as

(7) | |||

(8) |

where is the complementary Modulus.

The integral

(9) |

(10) | |||

(11) |

to write

(12) |

so

(13) |

(14) |

(15) |

(16) |

(17) |

(18) |

so

(19) |

(20) |

Therefore, we have proven the identity

(21) |

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

(22) | |||

(23) | |||

(24) | |||

(25) | |||

(26) | |||

(27) |

where

(28) |

(29) |

The Derivative of is

(30) |

(31) |

(32) |

**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. 61-62 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