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In integral calculus, an elliptic integral is any function f which can be expressed in the form
- <math> f(x) = \int_{c}^{x} R(t,P(t))\ dt <math>
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).
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 thier different appearence. Most texts adhere to a canonical naming scheme. Before defining the integrals, we review the naming conventions for the arguments:
- k the elliptic modulus
- m=k2 the parameter
- <math>\alpha<math> the modular angle, <math>k=\sin \alpha<math>
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.
Incomplete elliptic integral of the first kind
The incomplete elliptic integral of the first kind F is defined, in Jacobi's form, as
- <math> F(x;k) =
\int_{0}^{x} \frac{1}{ \sqrt{(1-t^2)(1-k^2 t^2)} }\ dt<math>
Equivalently, using alternate notation,
- <math> F(x;k) = F(\phi|m) = F(\phi\setminus \alpha ) =
\int_0^\phi \frac{1}{ \sqrt{1-\sin^2 \alpha \sin^2 \theta}} \ d\theta
<math>
where it is understood that when there is a verical 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. Note that
- <math>F(x;k) = u<math>
with u as defined above: thus, the jacobian elliptic functions are inverses to the elliptic integrals.
Incomplete elliptic integral of the second kind
The incomplete elliptic integral of the second kind E is
- <math> E(x;k) = \int_{0}^{x} \frac{ \sqrt{1-k^2 t^2} }{ \sqrt{1-t^2} }\ dt <math>
Equivalently, using alternate notation,
- <math> E(x;k) = E(\phi|m) = E(\phi\setminus \alpha ) =
\int_0^\phi \sqrt{1-\sin^2 \alpha \sin^2 \theta} \ d\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>
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}^{x} \frac{1}{1-nt^2}
\frac{1} {\sqrt{(1-k^2 t^2)(1-t^2) }}\ dt <math>
or
- <math> \Pi(n; \phi|m) = \int_0^\phi \frac{1}{1-n\sin^2 \theta}
\frac {1}{\sqrt{ (1-\sin^2 \alpha \sin^2 \theta) }} \ d\theta<math>
or
- <math> \Pi(n; \phi|m) = \int_0^u \frac{1}{1-n \textrm{sn}^2 (w|m)} \; dw<math>
The number n is called the characteristic and can take on any value, independently of the other arguments.
Complete elliptic integral of the first kind
The complete elliptic integral of the first kind K is defined as
- <math> K(k) = \int_{0}^{1} \frac{1}{ \sqrt{(1-t^2)(1-k^2 t^2)} }\ dt <math>
and can be computed in terms of the arithmetic-geometric mean.
It can also be calculated as
- <math> K(k) = \frac{\pi}{2} \sum_{n=0}^{\infty} k^{2n} \frac{(2n)!(2n)!}{16^n n!n!n!n!}<math>
Or in form of integral of sine, when 0 ≤ k ≤ 1
- <math>K( k ) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt {1 - k^2 \sin ^2 \theta }}<math>
The complete elliptic integral of the first kind is sometimes called he quarter period.
Complete elliptic integral of the second kind
The complete elliptic integral of the second kind E is defined as
- <math> E(k) = \int_{0}^{1} \frac{ \sqrt{1-k^2 t^2} }{ \sqrt{1-t^2} }\ dt <math>
Or if 0 ≤ k ≤ 1:
- <math>E( k ) = \int_0^{\frac{\pi}{2}} \sqrt {1 - k^2 \sin ^2 \theta} d\theta<math>
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.
Historically, properties of these integrals were studied in connection with the problem of the arc length of an ellipse, by Fagnano and Leonhard Euler.
See also
References
- Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, (1964) Dover Publications, New York. ISBN 486-61272-4 . See chapter 17.
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