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 functions (§22.2); 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)}.

ⓘ

Defines:: $\operatorname{cs}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/22.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

… ►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 functions (§20.1) …

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 Generalizations and Analogs

… ►

§20.11(ii) Ramanujan’s Theta Function and $q$ -Series

… ► … ►

§20.11(iv) Theta Functions with Characteristics

… ►A further development on the lines of Neville’s notation (§20.1) is as follows. …

5: 20.7 Identities

… ►

§20.7(i) Sums of Squares

… ►

§20.7(ii) Addition Formulas

… ►

§20.7(v) Watson’s Identities

… ►

§20.7(vi) Landen Transformations

… ►

§20.7(vii) Derivatives of Ratios of Theta Functions

…