# elliptic integrals and Jacobi elliptic functions

For a modulus $0 (while here, we define the complementary modulus to $k$ to be the positive number $k^{\prime}$ with $k^{2}+k^{\prime 2}=1$) , write

 $\displaystyle F(\phi,k)$ $\displaystyle=$ $\displaystyle\int_{0}^{\phi}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}$ (1) $\displaystyle E(\phi,k)$ $\displaystyle=$ $\displaystyle\int_{0}^{\phi}\sqrt{1-k^{2}\sin^{2}\theta}\,d\theta$ (2) $\displaystyle\Pi(n,\phi,k)$ $\displaystyle=$ $\displaystyle\int_{0}^{\phi}\frac{d\theta}{(1+n\sin^{2}\theta)\sqrt{1-k^{2}% \sin^{2}\theta}}$ (3)

The change of variable $x=\sin\phi$ turns these into

 $\displaystyle F_{1}(x,k)$ $\displaystyle=$ $\displaystyle\int_{0}^{x}\frac{dv}{\sqrt{(1-v^{2})(1-k^{2}v^{2})}}$ (4) $\displaystyle E_{1}(x,k)$ $\displaystyle=$ $\displaystyle\int_{0}^{x}\sqrt{\frac{1-k^{2}v^{2}}{1-v^{2}}}\,dv$ (5) $\displaystyle\Pi_{1}(n,x,k)$ $\displaystyle=$ $\displaystyle\int_{0}^{x}\frac{dv}{(1+nv^{2})\sqrt{(1-v^{2})(1-k^{2}v^{2})}}$ (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 $\phi=\pi/2$, 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

 $\int\frac{q(x,y)}{r(x,y)}\,dx$

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 $\phi$ as a function of $F$, or vice versa. The notation used is

 $\phi=\mathrm{am}\,u\qquad u=\mathrm{arg}\,\phi$

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

 $\displaystyle\mathrm{sn}\,u$ $\displaystyle=$ $\displaystyle x$ $\displaystyle\mathrm{cn}\,u$ $\displaystyle=$ $\displaystyle\sqrt{1-x^{2}}$ $\displaystyle\mathrm{tn}\,u$ $\displaystyle=$ $\displaystyle\frac{\mathrm{sn}\,u}{\mathrm{cn}\,u}$ $\displaystyle\mathrm{dn}\,u$ $\displaystyle=$ $\displaystyle\sqrt{1-k^{2}x^{2}}$

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