Elliptic integrals

For a modulus $0<k<1$ (while here, we define the complementary modulus to $k$ to be the positive number $k'$ with $k^2+k'^2=1$ ) , write \begin{eqnarray} F(\phi,k)&=&\int_0^\phi\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}} \\ E(\phi,k)&=&\int_0^\phi\sqrt{1-k^2\sin^2\theta}\,d\theta \\ \Pi(n,\phi,k)&=&\int_0^\phi\frac{d\theta}{(1+n\sin^2\theta)\sqrt{1-k^2\sin^2\theta}} \end{eqnarray}The change of variable $x=\sin\phi$ turns these into \begin{eqnarray} F_1(x,k)&=&\int_0^x\frac{dv}{\sqrt{(1-v^2)(1-k^2v^2)}} \\ E_1(x,k)&=&\int_0^x\sqrt{\frac{1-k^2v^2}{1-v^2}}\,dv \\ \Pi_1(n,x,k)&=&\int_0^x\frac{dv}{(1+nv^2)\sqrt{(1-v^2)(1-k^2v^2)}} \end{eqnarray}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: \begin{eqnarray*} \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^2x^2} \end{eqnarray*} 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.