Elliptic integral
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 with no repeated roots, and c is a constant.
Particular examples include:
- 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 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>
- The incomplete elliptic integral of the first kind F is defined, in Jacobi's form, as
- <math> F(u;k) = \int_{0}^{u} \frac{1}{ \sqrt{(1-t^2)(1-k^2 t^2)} }\ dt <math>
- Likewise, the incomplete elliptic integral of the second kind E is
- <math> E(u;k) = \int_{0}^{u} \frac{ \sqrt{1-k^2 t^2} }{ \sqrt{1-t^2} }\ dt <math>
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.
The origin of the name
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:
