# elliptic integrals and Jacobi elliptic functions

For a modulus^{} $$ (while here, we define the *complementary modulus* to $k$ to be the positive number ${k}^{\prime}$ with ${k}^{2}+{k}^{\prime 2}=1$) , write

$F(\varphi ,k)$ | $=$ | $\int}_{0}^{\varphi}}{\displaystyle \frac{d\theta}{\sqrt{1-{k}^{2}{\mathrm{sin}}^{2}\theta}$ | (1) | ||

$E(\varphi ,k)$ | $=$ | ${\int}_{0}^{\varphi}}\sqrt{1-{k}^{2}{\mathrm{sin}}^{2}\theta}\mathit{d}\theta $ | (2) | ||

$\mathrm{\Pi}(n,\varphi ,k)$ | $=$ | $\int}_{0}^{\varphi}}{\displaystyle \frac{d\theta}{(1+n{\mathrm{sin}}^{2}\theta )\sqrt{1-{k}^{2}{\mathrm{sin}}^{2}\theta}$ | (3) |

The change of variable $x=\mathrm{sin}\varphi $ turns these into

${F}_{1}(x,k)$ | $=$ | $\int}_{0}^{x}}{\displaystyle \frac{dv}{\sqrt{(1-{v}^{2})(1-{k}^{2}{v}^{2})}$ | (4) | ||

${E}_{1}(x,k)$ | $=$ | ${\int}_{0}^{x}}\sqrt{{\displaystyle \frac{1-{k}^{2}{v}^{2}}{1-{v}^{2}}}}\mathit{d}v$ | (5) | ||

${\mathrm{\Pi}}_{1}(n,x,k)$ | $=$ | $\int}_{0}^{x}}{\displaystyle \frac{dv}{(1+n{v}^{2})\sqrt{(1-{v}^{2})(1-{k}^{2}{v}^{2})}$ | (6) |

The first three functions^{} are known as Legendre’s form of the incomplete
elliptic integrals of the first, second, and third kinds respectively.
Notice that (2) is the special case $n=0$ of (3).
The latter three are known as Jacobi’s form of those integrals^{}.
If $\varphi =\pi /2$, or $x=1$, they are called complete^{} rather than incomplete
integrals, and we refer to the auxiliary elliptic integrals $K(k)=F(\pi /2,k)$, $E(k)=E(\pi /2,k)$, etc.

One use for elliptic integrals is to systematize the evaluation of
certain other integrals.
In particular, let $p$ be a third- or fourth-degree polynomial^{}
in one variable, and let $y=\sqrt{p(x)}$.
If $q$ and $r$ are any two polynomials in two variables, then the
indefinite integral

$$\int \frac{q(x,y)}{r(x,y)}\mathit{d}x$$ |

has a “closed form^{}” in terms of the above incomplete elliptic integrals,
together with elementary functions^{} and their inverses.

Jacobi’s elliptic functions^{}

In (1) we may regard $\varphi $ as a function of $F$, or vice versa. The notation used is

$$\varphi =\mathrm{am}u\mathit{\hspace{1em}\hspace{1em}}u=\mathrm{arg}\varphi $$ |

and $\varphi $ and $u$ are known as the amplitude and argument^{} respectively.
But $x=\mathrm{sin}\varphi =\mathrm{sin}\mathrm{am}u$.
The function $u\mapsto \mathrm{sin}\mathrm{am}u=x$
is denoted by $\mathrm{sn}$ and is one of four *Jacobi (or Jacobian ^{})
elliptic functions*. The four are:

$\mathrm{sn}u$ | $=$ | $x$ | ||

$\mathrm{cn}u$ | $=$ | $\sqrt{1-{x}^{2}}$ | ||

$\mathrm{tn}u$ | $=$ | $\frac{\mathrm{sn}u}{\mathrm{cn}u}$ | ||

$\mathrm{dn}u$ | $=$ | $\sqrt{1-{k}^{2}{x}^{2}}$ |

When the Jacobian elliptic functions^{} are extended to complex arguments,
they are doubly periodic and have two poles in any parallelogram of
periods; both poles are simple.

Title | elliptic integrals and Jacobi elliptic functions |

Canonical name | EllipticIntegralsAndJacobiEllipticFunctions |

Date of creation | 2013-03-22 13:58:28 |

Last modified on | 2013-03-22 13:58:28 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 33E05 |

Related topic | ArithmeticGeometricMean |

Related topic | PerimeterOfEllipse |

Defines | elliptic integral |

Defines | Jacobi elliptic function |

Defines | Jacobian elliptic function |

Defines | complementary modulus |

Defines | complete elliptic integral |