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Neville theta functions

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1: 20.1 Special Notation
Neville’s notation: θ s ( z | τ ) , θ c ( z | τ ) , θ d ( z | τ ) , θ n ( z | τ ) , respectively, for θ 3 2 ( 0 | τ ) θ 1 ( u | τ ) / θ 1 ( 0 | τ ) , θ 2 ( u | τ ) / θ 2 ( 0 | τ ) , θ 3 ( u | τ ) / θ 3 ( 0 | τ ) , θ 4 ( u | τ ) / θ 4 ( 0 | τ ) , where again u = z / θ 3 2 ( 0 | τ ) . This notation simplifies the relationship of the theta functions to Jacobian elliptic functions22.2); see Neville (1951). …
2: 22.2 Definitions
22.2.9 sc ( z , k ) = θ 3 ( 0 , q ) θ 4 ( 0 , q ) θ 1 ( ζ , q ) θ 2 ( ζ , q ) = 1 cs ( z , k ) .
The six functions containing the letter 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 θ s ( x , q ) , θ c ( x , q ) , θ d ( x , q ) , θ n ( x , q ) (see §20.1) and their logarithmic x -derivatives are given in Abramowitz and Stegun (1964, pp. 582–585) to 9D for ε , α = 0 ( 5 ) 90 , where (in radian measure) ε = x / θ 3 2 ( 0 , q ) = π x / ( 2 K ( k ) ) , and α 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