Legendre form

In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because^[1] the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity ${\displaystyle \scriptstyle {k}}$ (the ellipse being defined parametrically by ${\displaystyle \scriptstyle {x={\sqrt {1-k^{2}}}\cos(t)}}$, ${\displaystyle \scriptstyle {y=\sin(t)}}$).

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.