## About Elliptic Integrals of the First Kind

### Definitions

The incomplete elliptic integral of the first kind is defined as:

\[ \textrm{F}(\varphi,k) = \int_{0}^{\varphi}\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}} \]

where k is the elliptic modulus, with \(-1 \le k \le 1 \). Variable \(\varphi \) is the Jacobi's amplitude.

The complete elliptic integral of the first kind is defined as:

\[ \textrm{K}(k) = \textrm{F}(\frac{\pi}{2},k) = \int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}} \]

### Values

In the following table the values of the complete elliptic integral of the first kind are shown for a range of k values:

k | K(k) |
---|---|

-1 | |

-0.999 | |

-0.99 | |

-0.98 | |

-0.97 | |

-0.96 | |

-0.95 | |

-0.9 | |

-0.8 | |

-0.7 | |

-0.6 | |

-0.5 | |

-0.4 | |

-0.3 | |

-0.2 | |

-0.1 | |

0 | |

0.1 | |

0.2 | |

0.3 | |

0.4 | |

0.5 | |

0.6 | |

0.7 | |

0.8 | |

0.9 | |

0.95 | |

0.96 | |

0.97 | |

0.98 | |

0.99 | |

0.999 | |

1 |

In the following table the values of the incomplete elliptic integral of the first kind are shown for a range of k and φ values:

k | F(30°,k) | F(45°,k) | F(60°,k) | F(90°,k) | F(135°,k) | F(225°,k) | F(270°,k) | F(315°,k) |
---|---|---|---|---|---|---|---|---|

-1 | ||||||||

-0.999 | ||||||||

-0.99 | ||||||||

-0.98 | ||||||||

-0.97 | ||||||||

-0.96 | ||||||||

-0.95 | ||||||||

-0.9 | ||||||||

-0.8 | ||||||||

-0.7 | ||||||||

-0.6 | ||||||||

-0.5 | ||||||||

-0.4 | ||||||||

-0.3 | ||||||||

-0.2 | ||||||||

-0.1 | ||||||||

0 | ||||||||

0.1 | ||||||||

0.2 | ||||||||

0.3 | ||||||||

0.4 | ||||||||

0.5 | ||||||||

0.6 | ||||||||

0.7 | ||||||||

0.8 | ||||||||

0.9 | ||||||||

0.95 | ||||||||

0.96 | ||||||||

0.97 | ||||||||

0.98 | ||||||||

0.99 | ||||||||

0.999 | ||||||||

1 |

The xy plots of the incomplete integral of the first kind for various values of amplitude φ are depicted in the following figure: