This may be a question whose answer is lost in the mists of time, but why is the elliptical integral of the first kind denoted as $F(\pi/2,m)=K(m)$ when that of the second kind has $E(\pi/2,m)=E(m)$? It's not very consistent! Aside from convention, is there anything stopping us from rationalising these names a little?
Inconsistent naming of elliptic integrals
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soft-question
notation
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0According to Cajori's book, anyway, we have, as expected, Legendre and Jacobi to blame for using $K$ in the complete case and $F$ in the incomplete case for the elliptic integral of the first kind. I have no access to those 19th century articles where they were first introduced, so my guess of why they chose these letters is as good as yours. – 2010-11-05
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This discrepancy seems to come from the various conventions in defining the nome $q$ of complex lattice $\Lambda_{\tau} = \mathbb{Z} \oplus \tau \mathbb{Z}$ as one of the four $e^{2 \pi i \tau}$, $e^{\pi i \tau}$, $e^{2 i \tau}$ or $e^{i \tau}$, where $\tau \in \mathbb{H}$ is the lattice parameter. Investigate the vast literature on theta functions, and you'll see exactly what I mean by the problem of too many "conventions".
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0The literature of elliptic integrals/elliptic functions is a notational mess of its own. Note the very big list of notations at the top of [this page](http://reference.wolfram.com/mathematica/tutorial/EllipticIntegralsAndEllipticFunctions.html). – 2010-11-05