# §20.15 Tables

Theta functions are tabulated in Jahnke and Emde (1945, p. 45). This reference gives $\theta_{j}\left(x,q\right)$, $j=1,2,3,4$, and their logarithmic $x$-derivatives to 4D for $x/\pi=0(.1)1$, $\alpha=0(9^{\circ})90^{\circ}$, where $\alpha$ is the modular angle given by

 20.15.1 $\sin\alpha={\theta_{2}^{2}}\left(0,q\right)/{\theta_{3}^{2}}\left(0,q\right)=k.$ ⓘ Defines: $\alpha$: modular angle (locally) Symbols: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$: theta function, $\sin\NVar{z}$: sine function and $q$: nome Referenced by: §20.15, §20.15 Permalink: http://dlmf.nist.gov/20.15.E1 Encodings: TeX, pMML, png See also: Annotations for §20.15 and Ch.20

Spenceley and Spenceley (1947) tabulates $\theta_{1}\left(x,q\right)/\theta_{2}\left(0,q\right)$, $\theta_{2}\left(x,q\right)/\theta_{2}\left(0,q\right)$, $\theta_{3}\left(x,q\right)/\theta_{4}\left(0,q\right)$, $\theta_{4}\left(x,q\right)/\theta_{4}\left(0,q\right)$ to 12D for $u=0(1^{\circ})90^{\circ}$, $\alpha=0(1^{\circ})89^{\circ}$, where $u=2x/(\pi{\theta_{3}^{2}}\left(0,q\right))$ and $\alpha$ is defined by (20.15.1), together with the corresponding values of $\theta_{2}\left(0,q\right)$ and $\theta_{4}\left(0,q\right)$.

Lawden (1989, pp. 270–279) tabulates $\theta_{j}\left(x,q\right)$, $j=1,2,3,4$, to 5D for $x=0(1^{\circ})90^{\circ}$, $q=0.1(.1)0.9$, and also $q$ to 5D for $k^{2}=0(.01)1$.

Tables of Neville’s theta functions $\theta_{s}\left(x,q\right)$, $\theta_{c}\left(x,q\right)$, $\theta_{d}\left(x,q\right)$, $\theta_{n}\left(x,q\right)$ (see §20.1) and their logarithmic $x$-derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for $\varepsilon,\alpha=0(5^{\circ})90^{\circ}$, where (in radian measure) $\varepsilon=x/{\theta_{3}^{2}}\left(0,q\right)=\pi x/(2\!K\left(k\right))$, and $\alpha$ is defined by (20.15.1).

For other tables prior to 1961 see Fletcher et al. (1962, pp. 508–514) and Lebedev and Fedorova (1960, pp. 227–230).