§20.1 Special Notation

(For other notation see Notation for the Special Functions.)

 $m$, $n$ integers. the argument. the lattice parameter, $\Im\tau>0$. the nome, $q=e^{i\pi\tau}$, $0<\left|q\right|<1$. Since $\tau$ is not a single-valued function of $q$, it is assumed that $\tau$ is known, even when $q$ is specified. Most applications concern the rectangular case $\Re\tau=0$, $\Im\tau>0$, so that $0 and $\tau$ and $q$ are uniquely related. $e^{i\alpha\pi\tau}$ for $\alpha\in\mathbb{R}$ (resolving issues of choice of branch). set of all elements of $S_{1}$, modulo elements of $S_{2}$. Thus two elements of $S_{1}/S_{2}$ are equivalent if they are both in $S_{1}$ and their difference is in $S_{2}$. (For an example see §20.12(ii).)

The main functions treated in this chapter are the theta functions $\theta_{j}\left(z\middle|\tau\right)=\theta_{j}\left(z,q\right)$ where $j=1,2,3,4$ and $q=e^{i\pi\tau}$. When $\tau$ is fixed the notation is often abbreviated in the literature as $\theta_{j}(z)$, or even as simply $\theta_{j}$, it being then understood that the argument is the primary variable. Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21.

Primes on the $\theta$ symbols indicate derivatives with respect to the argument of the $\theta$ function.

Other Notations

Jacobi’s original notation: $\Theta(z|\tau)$, $\Theta_{1}(z|\tau)$, $\mathrm{H}(z|\tau)$, $\mathrm{H}_{1}(z|\tau)$, respectively, for $\theta_{4}\left(u\middle|\tau\right)$, $\theta_{3}\left(u\middle|\tau\right)$, $\theta_{1}\left(u\middle|\tau\right)$, $\theta_{2}\left(u\middle|\tau\right)$, where $u=z/{\theta_{3}^{2}}\left(0\middle|\tau\right)$. Here the symbol $\mathrm{H}$ denotes capital eta. See, for example, Whittaker and Watson (1927, p. 479) and Copson (1935, pp. 405, 411).

Neville’s notation: $\theta_{s}(z|\tau)$, $\theta_{c}(z|\tau)$, $\theta_{d}(z|\tau)$, $\theta_{n}(z|\tau)$, respectively, for ${\theta_{3}^{2}}\left(0\middle|\tau\right)\ifrac{\theta_{1}\left(u\middle|\tau% \right)}{\theta_{1}'\left(0\middle|\tau\right)}$, $\ifrac{\theta_{2}\left(u\middle|\tau\right)}{\theta_{2}\left(0\middle|\tau% \right)}$, $\ifrac{\theta_{3}\left(u\middle|\tau\right)}{\theta_{3}\left(0\middle|\tau% \right)}$, $\ifrac{\theta_{4}\left(u\middle|\tau\right)}{\theta_{4}\left(0\middle|\tau% \right)}$, where again $u=z/{\theta_{3}^{2}}\left(0\middle|\tau\right)$. This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951).

McKean and Moll’s notation: $\vartheta_{j}(z|\tau)=\theta_{j}\left(\pi z\middle|\tau\right)$, $j=1,2,3,4$. See McKean and Moll (1999, p. 125).

Additional notations that have been used in the literature are summarized in Whittaker and Watson (1927, p. 487).