PlanetMath: elliptic integrals and Jacobi elliptic functions

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elliptic integrals and Jacobi elliptic functions

(Definition)

Elliptic integrals

For a modulus (while here, we define the complementary modulus to to be the positive number with ) , 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

$\displaystyle F_1(x,k)$	$\displaystyle =$	$\displaystyle \int_0^x\frac{dv}{\sqrt{(1-v^2)(1-k^2v^2)}}$	(4)
$\displaystyle E_1(x,k)$	$\displaystyle =$	$\displaystyle \int_0^x\sqrt{\frac{1-k^2v^2}{1-v^2}}\,dv$	(5)
$\displaystyle \Pi_1(n,x,k)$	$\displaystyle =$	$\displaystyle \int_0^x\frac{dv}{(1+nv^2)\sqrt{(1-v^2)(1-k^2v^2)}}$	(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

of (3). The latter three are known as Jacobi's form of those integrals. If $\phi=\pi/2$ , or

, 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 be a third- or fourth-degree polynomial in one variable, and let $y=\sqrt{p(x)}$ . If and are any two polynomials in two variables, then the indefinite integral

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

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 , or vice versa. The notation used is

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

and $\phi$ and

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:

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

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.

"elliptic integrals and Jacobi elliptic functions" is owned by mathcam. [ full author list (3) | owner history (1) ]

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Also defines:

elliptic integral, Jacobi elliptic function, Jacobian elliptic function, complementary modulus, complete elliptic integral

Attachments:: reduction of elliptic integrals to standard form (Theorem) by rspuzio

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Cross-references: parallelogram, poles, doubly periodic, elliptic functions, Jacobian, argument, inverses, elementary functions, indefinite integral, polynomial, integrals, functions, variable, number, positive, modulus
There are 10 references to this entry.

This is version 4 of elliptic integrals and Jacobi elliptic functions, born on 2003-09-30, modified 2006-12-10.
Object id is 4746, canonical name is EllipticIntegralsAndJacobiEllipticFunctions.
Accessed 14341 times total.

Classification:

AMS MSC:

33E05 (Special functions :: Other special functions :: Elliptic functions and integrals)

Pending Errata and Addenda

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