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I am doing some calculation involving elliptic integrals/functions, and find the notations confusing.

In Wittaker-Watson, the "Jacobi's earlier notation" H(u) is called the Eta-function, so the "H" is not the Latin letter eitch but the Greek capital Eta. Similarly the function Z(u), defined as $\mathrm{Z}(u) = \Theta'(u)/\Theta(u)$, is called the Zeta function, so the "Z" is not the Latin letter zed but the Greek Zeta.

My question is:

(1) What is the notation for E(u) that is related to $\mathrm{Z}(u)$ as E(u) = Z(u) + uE/K? Is it the Latin e, or the Greek capital Epsilon?

(2) The three kinds of elliptic integrals, in Legendre's form, are denoted as $F, E, \Pi$ respectively. I wonder why Latin and Greek notations are mixed. It seems that $\Pi$ have to be Greek, and $F$ have to be Latin. But how about E, it is e or Eta?

I am asking this seemingly trivial question because in common $\LaTeX$ typesetting, capital Latin letters are italic and capital Greek letters are roman (like $Z$ and $\mathrm{Z}$). Thus I want to distinguish them in writing. Whittaker-Watson does not help in this aspect, since all notations are italic in this old book.

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Wait, is this why the copy editor changed every slanted Z in $Z_n$ to the upright version in my paper? I'm starting to be curious if the letter Z for the ring $Z_n$ (i.e., $Z/nZ$) is Latin, Greek or something else... I never understood the difference between italic and slanted either. – Yuichiro Fujiwara Jan 4 '13 at 5:10
up vote 2 down vote accepted

According to Whittaker Watson, these notations were introduced in Jacobi's paper De functionibus ellipticis commentario, J. fur Math., 1829, IV, p. 371. This paper is easily available on line. In it, $E$ is slanted (italic), while $\Pi, \Theta$ and $\Delta$ are not. I conclude that Jacobi either meant Latin $E$, or simply did not care:-)

Both TeX and LaTeX say that \Epsilon and \Eta are "undefined commands". I suspect it was the same in Jacobi's time:-)

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