Legendre's relation

In mathematics, Legendre's relation can be expressed in either of two forms: as a relation between complete elliptic integrals, or as a relation between periods and quasiperiods of elliptic functions. The two forms are equivalent as the periods and quasiperiods can be expressed in terms of complete elliptic integrals. It was introduced (for complete elliptic integrals) by A. M. Legendre (1811, 1825, p. 61).

Complete elliptic integrals

Legendre's relation stated using complete elliptic integrals is

K'E+KE'-KK'={\frac {\pi }{2}}

where K and K′ are the complete elliptic integrals of the first kind for values satisfying k² + k′² = 1, and E and E′ are the complete elliptic integrals of the second kind.

This form of Legendre's relation expresses the fact that the Wronskian of the complete elliptic integrals (considered as solutions of a differential equation) is a constant.

Elliptic functions

Legendre's relation stated using elliptic functions is

\omega _{2}\eta _{1}-\omega _{1}\eta _{2}=2\pi i\,

where ω₁ and ω₂ are the periods of the Weierstrass elliptic function, and η₁ and η₂ are the quasiperiods of the Weierstrass zeta function. Some authors normalize these in a different way differing by factors of 2, in which case the right hand side of the Legendre relation is $π$ i or $π$ i / 2. This relation can be proved by integrating the Weierstrass zeta function about the boundary of a fundamental region and applying Cauchy's residue theorem.

References

Duren, Peter (1991), "The Legendre relation for elliptic integrals", in Ewing, John H.; Gehring, F. W. (eds.), Paul Halmos. Celebrating 50 years of mathematics, New York: Springer-Verlag, pp. 305-315, doi:10.1007/978-1-4612-0967-6_32, ISBN 0-387-97509-8, MR 1113282
Karatsuba, E. A.; Vuorinen, M. (2001), "On hypergeometric functions and generalizations of Legendre's relation", J. Math. Anal. Appl., 260 (2): 623–640, MR 1845572
Legendre, A.M. (1811), Exercices de calcul intégral sur divers ordres de transcendantes et sur les quadratures, vol. I, Paris
Legendre, A.M. (1825), Traité des fonctions elliptiques et des intégrales eulériennes, vol. I, Paris