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1: 22.2 Definitions
§22.2 Definitions
The nome q is given in terms of the modulus k by
k = θ 2 2 ( 0 , q ) θ 3 2 ( 0 , q ) ,
k = θ 4 2 ( 0 , q ) θ 3 2 ( 0 , q ) ,
2: 22.11 Fourier and Hyperbolic Series
Throughout this section q and ζ are defined as in §22.2. If q exp ( 2 | ζ | ) < 1 , then
22.11.1 sn ( z , k ) = 2 π K k n = 0 q n + 1 2 sin ( ( 2 n + 1 ) ζ ) 1 - q 2 n + 1 ,
Next, if q exp ( | ζ | ) < 1 , then … Next, with E = E ( k ) denoting the complete elliptic integral of the second kind (§19.2(ii)) and q exp ( 2 | ζ | ) < 1 , …
3: 20.4 Values at z = 0
20.4.2 θ 1 ( 0 , q ) = 2 q 1 / 4 n = 1 ( 1 - q 2 n ) 3 ,
20.4.3 θ 2 ( 0 , q ) = 2 q 1 / 4 n = 1 ( 1 - q 2 n ) ( 1 + q 2 n ) 2 ,
20.4.4 θ 3 ( 0 , q ) = n = 1 ( 1 - q 2 n ) ( 1 + q 2 n - 1 ) 2 ,
20.4.5 θ 4 ( 0 , q ) = n = 1 ( 1 - q 2 n ) ( 1 - q 2 n - 1 ) 2 .
20.4.6 θ 1 ( 0 , q ) = θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 4 ( 0 , q ) .
4: 19.5 Maclaurin and Related Expansions
For Jacobi’s nome q :
19.5.5 q = exp ( - π K ( k ) / K ( k ) ) = r + 8 r 2 + 84 r 3 + 992 r 4 + , r = 1 16 k 2 , 0 k 1 .
19.5.6 q = λ + 2 λ 5 + 15 λ 9 + 150 λ 13 + 1707 λ 17 + , 0 k 1 ,
19.5.8 K ( k ) = π 2 ( 1 + 2 n = 1 q n 2 ) 2 , | q | < 1 ,
5: 20.3 Graphics
§20.3(i) θ -Functions: Real Variable and Real Nome
§20.3(ii) θ -Functions: Complex Variable and Real Nome
6: 20.1 Special Notation
m , n integers.
q ( ) the nome, q = e i π τ , 0 < | q | < 1 . Since τ is not a single-valued function of q , it is assumed that τ is known, even when q is specified. Most applications concern the rectangular case τ = 0 , τ > 0 , so that 0 < q < 1 and τ and q are uniquely related.
q α e i α π τ for α (resolving issues of choice of branch).
7: 20.8 Watson’s Expansions
20.8.1 θ 2 ( 0 , q ) θ 3 ( z , q ) θ 4 ( z , q ) θ 2 ( z , q ) = 2 n = - ( - 1 ) n q n 2 e i 2 n z q - n e - i z + q n e i z .
8: 22.1 Special Notation
x , y real variables.
q nome. 0 q < 1 except in §22.17; see also §20.1.
9: 23.17 Elementary Properties
23.17.4 λ ( τ ) = 16 q ( 1 - 8 q + 44 q 2 + ) ,
23.17.5 1728 J ( τ ) = q - 2 + 744 + 1 96884 q 2 + 214 93760 q 4 + ,
23.17.6 η ( τ ) = n = - ( - 1 ) n q ( 6 n + 1 ) 2 / 12 .
23.17.7 λ ( τ ) = 16 q n = 1 ( 1 + q 2 n 1 + q 2 n - 1 ) 8 ,
23.17.8 η ( τ ) = q 1 / 12 n = 1 ( 1 - q 2 n ) ,
10: 23.15 Definitions
In §§23.1523.19, k and k ( ) denote the Jacobi modulus and complementary modulus, respectively, and q = e i π τ ( τ > 0 ) denotes the nome; compare §§20.1 and 22.1. …
23.15.6 λ ( τ ) = θ 2 4 ( 0 , q ) θ 3 4 ( 0 , q ) ;
23.15.7 J ( τ ) = ( θ 2 8 ( 0 , q ) + θ 3 8 ( 0 , q ) + θ 4 8 ( 0 , q ) ) 3 54 ( θ 1 ( 0 , q ) ) 8 ,
23.15.8 θ 1 ( 0 , q ) = θ 1 ( z , q ) / z | z = 0 .