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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?

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Of course you're free to use any names you wish for these functions, but the problem is that people might not understand what you mean. There have been attempts to clean up the notation, for example Eagle's book "The Elliptic Functions as They Should Be: An Account, with Applications, of the Functions in a New Canonical Form", but so far the traditional notation has prevailed (as usual). – Hans Lundmark Nov 5 '10 at 7:37
@HansLundmark Thanks, interesting reference. Considering the field I'm in still requires boilerplate like "where E(m) is the second complete elliptic integral" I'm guessing you could sneak in a slight notation change without causing too much angst. But we'll see! – Will Robertson Nov 5 '10 at 7:55
Eh, in my ideal world, we'll all be using Carlson's integrals, but the Legendre-Jacobi ones have a long tradition of use in applications, and I suppose we won't be getting rid of them anytime soon... – J. M. Nov 5 '10 at 11:08
According 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. – J. M. Nov 5 '10 at 11:17

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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The 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. – J. M. Nov 5 '10 at 14:04

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