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Landen's Transformation

If $x\sin\alpha =\sin(2\beta -\alpha)$, then

(1+x)\int_0^\alpha {d\phi\over \sqrt{1-x^2\sin^2\phi}}
= 2 \int_0^\beta {d\phi \over \sqrt{1-{4x\over (1+x)^2}\sin^2\phi }}.

See also Elliptic Integral of the First Kind, Gauss's Transformation


Abramowitz, M. and Stegun, C. A. (Eds.). ``Ascending Landen Transformation'' and ``Landen's Transformation.'' §16.14 and 17.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 573-574 and 597-598, 1972.

© 1996-9 Eric W. Weisstein