20 Theta Functions Notation 20 Theta Functions 20.2 Definitions and Periodic Properties

§20.1 Special Notation

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Keywords:: lattice parameter, nome, notation, rectangular case, theta functions
Referenced by:: §20.11(v), §20.15, §20.7(ii), §22.1, §22.2, §23.15(i)
Permalink:: http://dlmf.nist.gov/20.1
See also:: Annotations for Ch.20

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

$m$ , $n$	integers.
$z$ $(\in\mathbb{C})$	the argument.
$\tau$ $(\in\mathbb{C})$	the lattice parameter, $\Im\tau>0$ .
$q$ $(\in\mathbb{C})$	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 $0<q<1$ and $\tau$ and $q$ are uniquely related.
$q^{\alpha}$	$e^{i\alpha\pi\tau}$ 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 $\theta_{j}\left(z\middle|\tau\right)=\theta_{j}\left(z,q\right)$ 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

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Keywords:: Jacobi’s original notation, McKean and Moll’s, McKean and Moll’s theta functions, Neville’s, Neville’s theta functions, theta functions
See also:: Annotations for §20.1 and Ch.20

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).

20 Theta Functions 20.2 Definitions and Periodic Properties

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