§20.1 Special Notation

Keywords:: lattice parameter, nome, 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

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

$m$ , $n$	integers.
$z$ $(\in\Complex)$	the argument.
$\tau$ $(\in\Complex)$	the lattice parameter, $\imagpart{\tau}>0$ .
$q$ $(\in\Complex)$	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 $\realpart{\tau}=0$ , $\imagpart{\tau}>0$ , so that $0<q<1$ and $\tau$ and $q$ are uniquely related.
$q^{{\alpha}}$	$e^{{i\alpha\pi\tau}}$ for $\alpha\in\Real$ (resolving issues of choice of branch).
$S_{1}\setmod S_{2}$	set of all elements of $S_{1}$ , modulo elements of $S_{2}$ . Thus two elements of $S_{1}\setmod S_{2}$ 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 $\mathop{\theta _{{j}}\/}\nolimits\!\left(z\middle|\tau\right)=\mathop{\theta _{{j}}\/}\nolimits\!\left(z,q\right)$ 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 $\mathop{\theta _{{j}}\/}\nolimits(z)$ , or even as simply $\mathop{\theta _{{j}}\/}\nolimits$ , 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 $\mathop{\theta\/}\nolimits$ symbols indicate derivatives with respect to the argument of the $\mathop{\theta\/}\nolimits$ function.

¶ Other Notations

Keywords:: McKean and Moll’s theta functions, Neville’s theta functions, theta functions

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

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