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 + K E ′ − K K ′ = π 2 {\displaystyle 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
ω 2 η 1 − ω 1 η 2 = 2 π i {\displaystyle \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 πi or πi / 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.
Proof
Proof of the lemniscatic case The lemniscatic arc sine and the complementary lemniscatic arcsine are defined as follows:
arcsl ( r ) = ∫ 0 r 1 1 − ρ 4 d ρ = 1 2 2 K ( 1 2 2 ) − 1 2 2 E [ arccos ( r ) ; 1 2 2 ] {\displaystyle \operatorname {arcsl} (r)=\int _{0}^{r}{\frac {1}{\sqrt {1-\rho ^{4}}}}\,\mathrm {d} \rho ={\frac {1}{2}}{\sqrt {2}}\,K{\bigl (}{\tfrac {1}{2}}{\sqrt {2}}{\bigr )}-{\frac {1}{2}}{\sqrt {2}}\,E{\bigl [}\arccos(r);{\tfrac {1}{2}}{\sqrt {2}}{\bigr ]}}
arcsl ∗ ( r ) = ∫ 0 r 1 + ρ 2 1 − ρ 4 d ρ = 2 E ( 1 2 2 ) − 2 E [ arccos ( r ) ; 1 2 2 ] {\displaystyle \operatorname {arcsl} ^{*}(r)=\int _{0}^{r}{\frac {1+\rho ^{2}}{\sqrt {1-\rho ^{4}}}}\,\mathrm {d} \rho ={\sqrt {2}}\,E{\bigl (}{\tfrac {1}{2}}{\sqrt {2}}{\bigr )}-{\sqrt {2}}\,E{\bigl [}\arccos(r);{\tfrac {1}{2}}{\sqrt {2}}{\bigr ]}}
And these derivatives are valid:
d d r arcsl ( r ) = 1 1 − r 4 {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} r}}\operatorname {arcsl} (r)={\frac {1}{\sqrt {1-r^{4}}}}}
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