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

$m$, $n$ | integers. |
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$z$ $(\in \u2102)$ | the argument. |

$\tau $ $(\in \u2102)$ | the lattice parameter, $\mathrm{\Im}\tau >0$. |

$q$ $(\in \u2102)$ | the nome, $q={\mathrm{e}}^{\mathrm{i}\pi \tau}$, $$. Since $\tau $ is not a single-valued function of $q$, it is assumed that $\tau $ is known, even when $q$ is specified. Most applications concern the rectangular case $\mathrm{\Re}\tau =0$, $\mathrm{\Im}\tau >0$, so that $$ and $\tau $ and $q$ are uniquely related. |

${q}^{\alpha}$ | ${\mathrm{e}}^{\mathrm{i}\alpha \pi \tau}$ for $\alpha \in \mathbb{R}$ (resolving issues of choice of branch). |

${S}_{1}/{S}_{2}$ | 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 ${\theta}_{j}\left(z\right|\tau )={\theta}_{j}(z,q)$ where $j=1,2,3,4$ and $q={\mathrm{e}}^{\mathrm{i}\pi \tau}$. 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.

Jacobi’s original notation: $\mathrm{\Theta}(z|\tau )$, ${\mathrm{\Theta}}_{1}(z|\tau )$, $\mathrm{H}(z|\tau )$, ${\mathrm{H}}_{1}(z|\tau )$, respectively, for ${\theta}_{4}\left(u\right|\tau )$, ${\theta}_{3}\left(u\right|\tau )$, ${\theta}_{1}\left(u\right|\tau )$, ${\theta}_{2}\left(u\right|\tau )$, where $u=z/{\theta}_{3}^{2}\left(0\right|\tau )$. 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 ${\theta}_{3}^{2}\left(0\right|\tau ){\theta}_{1}\left(u\right|\tau )/{\theta}_{1}^{\prime}\left(0\right|\tau )$, ${\theta}_{2}\left(u\right|\tau )/{\theta}_{2}\left(0\right|\tau )$, ${\theta}_{3}\left(u\right|\tau )/{\theta}_{3}\left(0\right|\tau )$, ${\theta}_{4}\left(u\right|\tau )/{\theta}_{4}\left(0\right|\tau )$, where again $u=z/{\theta}_{3}^{2}\left(0\right|\tau )$. This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951).

McKean and Moll’s notation: ${\vartheta}_{j}(z|\tau )={\theta}_{j}\left(\pi z\right|\tau )$, $j=1,2,3,4$. 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).