# §20.15 Tables

Theta functions are tabulated in Jahnke and Emde (1945, p. 45). This reference gives $\mathop{\theta_{j}\/}\nolimits\!\left(x,q\right)$, $j=1,2,3,4$, 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 $\mathop{\sin\/}\nolimits\alpha={\mathop{\theta_{2}\/}\nolimits^{2}}\!\left(0,q% \right)/{\mathop{\theta_{3}\/}\nolimits^{2}}\!\left(0,q\right)=k.$ Defines: $\alpha$: modular angle (locally) Symbols: $\mathop{\theta_{\NVar{j}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$: theta function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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

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

Lawden (1989, pp. 270–279) tabulates $\mathop{\theta_{j}\/}\nolimits\!\left(x,q\right)$, $j=1,2,3,4$, 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 $\mathop{\theta_{s}\/}\nolimits\!\left(x,q\right)$, $\mathop{\theta_{c}\/}\nolimits\!\left(x,q\right)$, $\mathop{\theta_{d}\/}\nolimits\!\left(x,q\right)$, $\mathop{\theta_{n}\/}\nolimits\!\left(x,q\right)$ (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) $\varepsilon=x/{\mathop{\theta_{3}\/}\nolimits^{2}}\!\left(0,q\right)=\pi x/(2% \!\mathop{K\/}\nolimits\!\left(k\right))$, 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).