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Zbl 0787.33010
Duren, Peter
The Legendre relation for elliptic integrals. (English)
[A] Ewing, John H. (ed.) et al., Paul Halmos. Celebrating 50 years of mathematics. New York: Springer-Verlag. 305-315 (1991). ISBN 0-387-97509-8

Let $f,s,F$ and $S$ be the complete elliptic integrals of first or second kind, with parameter $k$ or $K=(1-k\sp 2)\sp{1/2}$, respectively. Legendre's relation asserts that $fS+Fs-fF=\pi/2$. Author presents three proofs. The first is by Legendre's method of evaluating elliptic integrals, the second via hypergeometric equations and the third based upon elliptic functions rather than integrals. While the author's favorite seems to be the third, this reviewer prefers the second: $f$ and $F$ are solutions of the same (hypergeometric) linear differential equation of second-order, so they satisfy a Wronskian relation which (after short but skillful calculations) turns out to be the Legendre relation.
[J.Aczél (Waterloo / Ontario)]

MSC 2000:: *33E05 Elliptic functions and integrals
33C05 Classical hypergeometric functions
33C60 Hypergeometric integrals and functions defined by them
33B15 Gamma-functions, etc.
34A30 Linear ODE and systems

Keywords: Jacobi function; Weierstrass function; Euler function; elliptic integrals; Wronskian relation; zeta function

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