# Neville theta functions

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##### 1: 20.1 Special Notation
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 functions22.2); see Neville (1951). …
##### 2: 22.2 Definitions
22.2.9 $\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{4}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q% \right)}=\frac{1}{\operatorname{cs}\left(z,k\right)}.$
The six functions containing the letter $\mathrm{s}$ in their two-letter name are odd in $z$; the other six are even in $z$. In terms of Neville’s theta functions20.1) …
##### 3: 20.15 Tables
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) $\varepsilon=x/{\theta_{3}}^{2}\left(0,q\right)=\pi x/(2K\left(k\right))$, and $\alpha$ is defined by (20.15.1). …
##### 4: 20.11 Generalizations and Analogs
###### §20.11(iv) ThetaFunctions with Characteristics
A further development on the lines of Neville’s notation (§20.1) is as follows. …