Elliptic integral

Definition

An elliptic integral is an integral involving a square root of a polynomial of degree 3 or 4. These integrals cannot generally be expressed in terms of elementary functions.

5 Must Know Facts For Your Next Test

1. Elliptic integrals are categorized into three types: the first kind, the second kind, and the third kind.
2. The incomplete elliptic integral of the first kind is denoted as $F(\phi, k)$ and involves parameters $\phi$ (amplitude) and $k$ (modulus).
3. The complete elliptic integral of the first kind is denoted as $K(k)$ and does not depend on $\phi$. It is defined as $K(k) = F(\frac{\pi}{2}, k)$.
4. Applications of elliptic integrals include calculating arc lengths of ellipses and solving Kepler’s problem in celestial mechanics.
5. Elliptic integrals can be approximated using Taylor series expansions for small values of the modulus $k$.

Review Questions

Related terms

Taylor Series:

A representation of a function as an infinite sum of terms calculated from its derivatives at a single point.

Arc Length:

The distance along a curve between two points.

Kepler's Problem: \text{A classical problem in celestial mechanics that involves determining the motion of planets based on gravitational forces.}