Elliptic integral

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In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler.

In the modern definition, an elliptic integral is any function f which can be expressed in the form

where R is a rational function of its two arguments, P is the square root of a polynomial of degree 3 or 4 (a cubic or quartic) with no repeated roots, and c is a constant.

In general, elliptic integrals cannot be expressed in terms of elementary functions; exceptions to this are when P does have repeated roots, or when R(x,y) contains no odd powers of y. However, with appropriate reduction formula, every elliptic integral can be brought into a form that involves integrals over rational functions, and the three canonical forms (i.e. the elliptic integrals of the first, second and third kind).

Besides the forms given below, the elliptic integrals may also be expressed in Legendre form and Carlson symmetric form. Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz-Christoffel mapping.

1 Notation
2 Incomplete elliptic integral of the first kind
3 Incomplete elliptic integral of the second kind
4 Incomplete elliptic integral of the third kind
5 Complete elliptic integral of the first kind
6 Complete elliptic integral of the second kind
7 History
8 See also
9 References

[edit] Notation

Elliptic integrals are often expressed as functions of a variety of different arguments. These different arguments are completely equivalent (they give the same elliptic integral), but can be confusing due to their different appearance. Most texts adhere to a canonical naming scheme. Before defining the integrals, we review the naming conventions for the arguments:

<math>o\!\varepsilon,\,\!</math> the modular angle;
<math>k=\sin o\!\varepsilon,\,\!</math> the elliptic modulus;
<math>m=k^2=\sin(o\!\varepsilon)^2,\,\!</math> the parameter;

Note that the above three are completely determined by one another; specifying one is the same as specifying another. The elliptic integrals will also depend on another argument; this can also be specified in a number of different ways:

<math>\phi\,\!</math> the amplitude
x where <math>x=\sin \phi= \textrm{sn} \; u\,\!</math>
u, where x = sn u and sn is one of the Jacobian elliptic functions

Specifying any one of these determines the others, and thus again, these may be used interchangeably in the notation. Note that u also depends on m. Some additional relationships involving u include

<math>\cos \phi = \textrm{cn}\; u\,\!</math>

and

<math>\sqrt{1-m\sin^2 \phi} = \textrm{dn}\; u.\,\!</math>

The latter is sometimes called the delta amplitude and written as <math>\Delta(\phi)=\textrm{dn}\; u\,\!</math>.

Sometimes the literature refers to the complementary parameter, the complementary modulus or the complementary modular angle. These are further defined in the article on quarter periods.

[edit] Incomplete elliptic integral of the first kind

The incomplete elliptic integral of the first kind F is defined as

<math> F(\phi\setminus o\!\varepsilon ) = F(\phi|m) =

\int_0^\phi\frac{d\theta}{\sqrt{1-(\sin\theta\sin o\!\varepsilon)^2}}\,\!</math>

Equivalently, using notation in Jacobi's form, one sets <math>x=\sin \phi ~,~ t=\sin \theta\;\!</math>; then

<math> F(\phi\setminus o\!\varepsilon ) = F(x;k) =

\int_{0}^{x} \frac{dt}{\sqrt{(1-t^2)(1-k^2 t^2)} }\,\!</math> where it is understood that when there is a vertical bar used, the argument following the vertical bar is the parameter (as defined above), and, when a backslash is used, it is followed by the modular angle. In this sense, <math> F(\sin\phi;\sin o\!\varepsilon) = F(\phi|\sin (o\!\varepsilon)^2) = F(\phi\setminus o\!\varepsilon )~ \,\!</math>. These notations are borrowed from the book Abramowitz and Stegun; the use of the delimiters ; | \ is traditional in elliptic integrals.

There are differing conventions regarding notation of elliptic integrals. The differences can be very confusing, especially to a novice, see [1]. The functions, which evaluate the elliptic integrals, do not have standard and unic names and meanings (like sqrt, sin and erf have). Different notations are used in the literature. Gradstein, Ryzhik [[2], Eq.(8.111)] and the wiki article Legendre form use notation <math> F(\phi,k) \,\!</math>, which is equivalent to our <math> F(\phi|k^2)~ \,\!</math>; also <math> E(\phi,k)=E(\phi|k^2)~ \,\!</math> below. For example, if one translates code from the language of Mathematica into the language of Maple, one should replace the argument of the EllipticK function by its square root, and correspondingly, in the translation from Maple to Mathematica the argument should be replaced by its square. EllipticK(x) in Maple is almost equivalent of EllipticK[x^2] in Mathematica; one may expect to get the same result in both systems, at least while 0<x<1.

Note that

with u as defined above: thus, the Jacobian elliptic functions are inverses to the elliptic integrals.

[edit] Incomplete elliptic integral of the second kind

The incomplete elliptic integral of the second kind E is

<math> E(\phi\setminus o\!\varepsilon) = E(\phi|m) =

\int_0^\phi\sqrt{1-(\sin\theta\sin o\!\varepsilon)^2}\ d\theta.\,\!</math>

Equivalently, using alternate notation (substituting <math>t=\sin\theta\,\!</math>),

Additional relations include

<math>E(\phi|m) = \int_0^u \textrm{dn}^2 w \;dw =

u-m\int_0^u \textrm{sn}^2 w \;dw = (1-m)u+m\int_0^u \textrm{cn}^2 w \;dw.\,\!</math>

[edit] Incomplete elliptic integral of the third kind

The incomplete elliptic integral of the third kind <math>\Pi\,\!</math> is

<math> \Pi(n; \phi|m) = \int_0^\phi \frac{1}{1-n\sin^2 \theta}

\frac {d\theta}{\sqrt{1-(\sin\theta\sin o\!\varepsilon)^2}},\,\!</math>

<math> \Pi(n; \phi|m) = \int_{0}^{x} \frac{1}{1-nt^2}

\frac{dt}{\sqrt{(1-k^2 t^2)(1-t^2) }},\,\!</math>

<math> \Pi(n; \phi|m) = \int_0^u \frac{dw}{1-n \textrm{sn}^2 (w|m)}. \; \,\!</math>

The number n is called the characteristic and can take on any value, independently of the other arguments. Note though that the value <math>\Pi(1;\pi/2|m)\,\!</math> is infinite, for any <math>m\,\!</math>.

[edit] Complete elliptic integral of the first kind

Main article: complete elliptic integral of the first kind.

[edit] Complete elliptic integral of the second kind

Main article: complete elliptic integral of the second kind.

[edit] History

Historically, elliptic functions were discovered as inverse functions of elliptic integrals, and this one in particular: we have F(sn(z;k);k) = z where sn is one of Jacobi's elliptic functions.