\scriptstyle{k}
\scriptstyle{x=\sqrt{1-k2
\scriptstyle{y=\sin(t)}
In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric forms. A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals.
The incomplete elliptic integral of the first kind is defined as,
F(\phi,k)=
| \phi | |
| \int | |
| 0 |
| 1 | |
| \sqrt{1-k2\sin2(t) |
the second kind as
E(\phi,k)=
| \phi | |
| \int | |
| 0 |
\sqrt{1-k2\sin2(t)}dt,
and the third kind as
\Pi(\phi,n,k)=
| \phi | |
| \int | |
| 0 |
| 1 | |
| (1-n\sin2(t))\sqrt{1-k2\sin2(t) |
The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign. The argument order shown above is that of Gradshteyn and Ryzhik[2] as well as Numerical Recipes.[3] The choice of sign is that of Abramowitz and Stegun as well as Gradshteyn and Ryzhik, but corresponds to the
\scriptstyle{\Pi(\phi,-n,k)}
The respective complete elliptic integrals are obtained by setting the amplitude,
\scriptstyle{\phi}
\scriptstyle{\pi/2}
The Legendre form of an elliptic curve is given by
y2=x(x-1)(x-λ)
The classic method of evaluation is by means of Landen's transformations. Descending Landen transformation decreases the modulus
\scriptstyle{k}
\scriptstyle{\phi}
\scriptstyle{k}
Most modern authors recommend evaluation in terms of the Carlson symmetric forms, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by Boost C++ Libraries, GNU Scientific Library and Numerical Recipes.[3]