Questions on elliptic integrals, integrals that involve the square root of a cubic or quartic polynomial.
An elliptic integral is most generally defined as $$\int R\left(t,\sqrt{P(t)}\right)\,dx$$ where $R$ is a rational function and $P$ is a cubic or quartic polynomial with no repeated roots. They arise in many fields of mathematics and physics.
Every elliptic integral may be expressed in terms of three standard forms (arguments follow Mathematica/mpmath conventions):
- The first kind: $$F(\varphi,m)=\int_0^\varphi\frac1{\sqrt{1-m\sin^2t}}\,dt$$
- The second kind: $$E(\varphi,m)=\int_0^\varphi\sqrt{1-m\sin^2t}\,dt$$
- The third kind: $$\Pi(n,\varphi,m)=\int_0^\varphi\frac1{(1-n\sin^2t)\sqrt{1-m\sin^2t}}\,dt$$
These incomplete integrals become complete when $\varphi=\frac\pi2$; their notations become $K(m),E(m)$ and $\Pi(n,m)$ respectively.
The inverse of $F(\varphi,m)$ for a fixed $m$ leads to the Jacobian elliptic-functions.
