Elliptic Integrals | Extreme Optimization Numerical Libraries for .NET Professional |

Elliptic integrals originally appeared in the calculation of arc lengths of ellipses. Later it was found that a larger class of integrals could be reduced to three kinds of elliptic integrals in Legendre form. The Special class provides an implementation of complete and incomplete elliptic integrals of the first, second and third kind.

Some ambiguity exists in the definition of these functions. For functions that take one parameter, the parameter is the elliptic modulus. For functions that takes two arguments, the additional parameter is the amplitude.

Method | Description |
---|---|

Complete elliptic integral of the first kind K(k). | |

Incomplete elliptic integral of the first kind F(φ,k). | |

Complete elliptic integral of the second kind E(k). | |

Incomplete elliptic integral of the second kind E(φ,k). | |

Complete elliptic integral of the third kind Π(n;k) with characteristic n. | |

Incomplete elliptic integral of the third kind Π(n;φ,k) with characteristic n. |

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