elliptic integrals and Jacobi elliptic functionsEllipticIntegralsAndJacobiEllipticFunctions2003-09-30 23:34:592006-12-10 13:40:23Definitionelliptic integralJacobi elliptic functionJacobian elliptic functioncomplementary moduluscomplete elliptic integral\usepackage{amssymb}
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\textbf{Elliptic integrals}
For a modulus $0<k<1$ (while here, we define the \emph{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.
\textbf{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 \emph{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.