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1: 20.9 Relations to Other Functions
§20.9(i) Elliptic Integrals
§20.9(ii) Elliptic Functions and Modular Functions
See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. The relations (20.9.1) and (20.9.2) between k and τ (or q ) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …
§20.9(iii) Riemann Zeta Function
2: 23.15 Definitions
§23.15 Definitions
3: 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 ) .
4: 20.3 Graphics
See accompanying text
Figure 20.3.2: θ 1 ( π x , q ) , 0 x 2 , q = 0. … Magnify
5: 21.2 Definitions
§21.2(iii) Relation to Classical Theta Functions
6: 19.10 Relations to Other Functions
§19.10(i) Theta and Elliptic Functions
For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. …
7: 22.16 Related Functions
Relation to Theta Functions
8: 19.25 Relations to Other Functions
§19.25(iv) Theta Functions
For relations of symmetric integrals to theta functions, see §20.9(i). …
9: 23.6 Relations to Other Functions
§23.6(i) Theta Functions
10: 20.11 Generalizations and Analogs
However, in this case q is no longer regarded as an independent complex variable within the unit circle, because k is related to the variable τ = τ ( k ) of the theta functions via (20.9.2). …