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The Legendre relation for elliptic integrals. (English) Zbl 0787.33010
Ewing, John H. (ed.) et al., Paul Halmos. Celebrating 50 years of mathematics. New York: Springer-Verlag. 305-315 (1991).
Let $f,s,F$ and $S$ be the complete elliptic integrals of first or second kind, with parameter $k$ or $K={\left(1-{k}^{2}\right)}^{1/2}$, 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.
MSC:
 33E05 Elliptic functions and integrals 33C05 Classical hypergeometric functions, ${}_{2}{F}_{1}$ 33C60 Hypergeometric integrals and functions defined by them 33B15 Gamma, beta and polygamma functions 34A30 Linear ODE and systems, general