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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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    $\begingroup$ 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. $\endgroup$
    – Yuichiro Fujiwara
    Commented Jan 4, 2013 at 5:10
  • $\begingroup$ @Yuichiro Z is for Zahlen, German for "numbers". $\endgroup$
    – David Roberts ♦
    Commented Sep 1, 2017 at 2:49
  • $\begingroup$ @YuichiroFujiwara ... was that copy-editor in Europe? In some conventions, mathematicians write italic letters for variables $a,b,c,x,y,z$ and roman letters for constants, such as $\mathrm{e}$ (Base of the natural logarithms) and $\mathrm i$ (${} = \sqrt{-1}$) and even $\mathrm{d}$ (differential operator as in $\int f(x)\;\mathrm{d}x$). But in the US we normally do not follow this convention for one-letter constants, writing $e,i,dx$. $\endgroup$
    – Gerald Edgar
    Commented Sep 1, 2017 at 10:59

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According to Whittaker and 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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