The Legendre Relation for Elliptic Integrals

Peter Duren
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Peter Duren: University of Michigan, Department of Mathematics

A chapter in PAUL HALMOS Celebrating 50 Years of Mathematics, 1991, pp 305-315 from Springer

Abstract: Abstract In standard notation, the complete elliptic integrals of the first and second kinds are $$ \begin{gathered} K = K(k) = \int_{0}^{{\pi /2}} {\frac{{d\theta }}{{\sqrt {{1 - {{k}^{2}}{{{\sin }}^{2}}\theta }} }}} \hfill \\ = \int_{0}^{1} {\frac{{dt}}{{\sqrt {{1 - {{t}^{2}}}} \sqrt {{1 - {{k}^{2}}{{t}^{2}}}} }}} \hfill \\ \end{gathered} $$ and $$ \begin{gathered} E = E(k) = \int_{0}^{{\pi /2}} {\sqrt {{1 - {{k}^{2}}{{{\sin }}^{2}}\theta d\theta }} } \hfill \\ = \int_{0}^{1} {\frac{{\sqrt {{1 - {{k}^{2}}{{t}^{2}}}} }}{{\sqrt {{1 - {{t}^{2}}}} }}dt.} \hfill \\ \end{gathered} $$

Keywords: Elliptic Function; Theta Function; Elliptic Integral; Complete Elliptic Integral; Distinct Complex Number (search for similar items in EconPapers)
Date: 1991
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DOI: 10.1007/978-1-4612-0967-6_32

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