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11—20 of 25 matching pages

11: 20.7 Identities
20.7.1 θ 3 2 ( 0 , q ) θ 3 2 ( z , q ) = θ 4 2 ( 0 , q ) θ 4 2 ( z , q ) + θ 2 2 ( 0 , q ) θ 2 2 ( z , q ) ,
20.7.2 θ 3 2 ( 0 , q ) θ 4 2 ( z , q ) = θ 2 2 ( 0 , q ) θ 1 2 ( z , q ) + θ 4 2 ( 0 , q ) θ 3 2 ( z , q ) ,
20.7.5 θ 3 4 ( 0 , q ) = θ 2 4 ( 0 , q ) + θ 4 4 ( 0 , q ) .
§20.7(iv) Reduction Formulas for Products
§20.7(ix) Addendum to 20.7(iv) Reduction Formulas for Products
12: 20.5 Infinite Products and Related Results
20.5.1 θ 1 ( z , q ) = 2 q 1 / 4 sin z n = 1 ( 1 - q 2 n ) ( 1 - 2 q 2 n cos ( 2 z ) + q 4 n ) ,
20.5.2 θ 2 ( z , q ) = 2 q 1 / 4 cos z n = 1 ( 1 - q 2 n ) ( 1 + 2 q 2 n cos ( 2 z ) + q 4 n ) ,
20.5.3 θ 3 ( z , q ) = n = 1 ( 1 - q 2 n ) ( 1 + 2 q 2 n - 1 cos ( 2 z ) + q 4 n - 2 ) ,
20.5.4 θ 4 ( z , q ) = n = 1 ( 1 - q 2 n ) ( 1 - 2 q 2 n - 1 cos ( 2 z ) + q 4 n - 2 ) .
20.5.9 θ 3 ( π z | τ ) = n = - p 2 n q n 2 = n = 1 ( 1 - q 2 n ) ( 1 + q 2 n - 1 p 2 ) ( 1 + q 2 n - 1 p - 2 ) ,
13: 23.1 Special Notation
𝕃 lattice in .
= e i π τ nome.
14: 23.12 Asymptotic Approximations
23.12.2 ζ ( z ) = π 2 4 ω 1 2 ( z 3 + 2 ω 1 π cot ( π z 2 ω 1 ) - 8 ( z - ω 1 π sin ( π z ω 1 ) ) q 2 + O ( q 4 ) ) ,
23.12.3 σ ( z ) = 2 ω 1 π exp ( π 2 z 2 24 ω 1 2 ) sin ( π z 2 ω 1 ) ( 1 - ( π 2 z 2 ω 1 2 - 4 sin 2 ( π z 2 ω 1 ) ) q 2 + O ( q 4 ) ) ,
23.12.4 η 1 = π 2 4 ω 1 ( 1 3 - 8 q 2 + O ( q 4 ) ) ,
15: 20.2 Definitions and Periodic Properties
20.2.1 θ 1 ( z | τ ) = θ 1 ( z , q ) = 2 n = 0 ( - 1 ) n q ( n + 1 2 ) 2 sin ( ( 2 n + 1 ) z ) ,
20.2.2 θ 2 ( z | τ ) = θ 2 ( z , q ) = 2 n = 0 q ( n + 1 2 ) 2 cos ( ( 2 n + 1 ) z ) ,
20.2.3 θ 3 ( z | τ ) = θ 3 ( z , q ) = 1 + 2 n = 1 q n 2 cos ( 2 n z ) ,
20.2.4 θ 4 ( z | τ ) = θ 4 ( z , q ) = 1 + 2 n = 1 ( - 1 ) n q n 2 cos ( 2 n z ) .
20.2.6 θ 1 ( z + ( m + n τ ) π | τ ) = ( - 1 ) m + n q - n 2 e - 2 i n z θ 1 ( z | τ ) ,
16: 20.11 Generalizations and Analogs
20.11.6 φ m , 1 ( z , q ) = θ 1 ( 0 , q ) θ m ( z , q ) θ m ( 0 , q ) θ 1 ( z , q ) , m = 2 , 3 , 4 ,
20.11.7 φ 1 , n ( z , q ) = θ n ( 0 , q ) θ 1 ( z , q ) θ 1 ( 0 , q ) θ n ( z , q ) , n = 2 , 3 , 4 ,
20.11.8 φ m , n ( z , q ) = θ n ( 0 , q ) θ m ( z , q ) θ m ( 0 , q ) θ n ( z , q ) , m , n = 2 , 3 , 4 .
20.11.9 φ m , n ( z , q ) = φ m , 1 ( z , q ) φ 1 , n ( z , q ) = 1 φ n , m ( z , q ) = φ m , 1 ( z , q ) φ n , 1 ( z , q ) = φ 1 , n ( z , q ) φ 1 , m ( z , q ) .
17: 22.21 Tables
Curtis (1964b) tabulates sn ( m K / n , k ) , cn ( m K / n , k ) , dn ( m K / n , k ) for n = 2 ( 1 ) 15 , m = 1 ( 1 ) n - 1 , and q (not k ) = 0 ( .005 ) 0.35 to 20D. …
18: 22.16 Related Functions
With q as in (22.2.1) and ζ = π x / ( 2 K ) ,
22.16.9 am ( x , k ) = π 2 K x + 2 n = 1 q n sin ( 2 n ζ ) n ( 1 + q 2 n ) .
where ξ = x / θ 3 2 ( 0 , q ) . …
19: 22.20 Methods of Computation
If either τ or q = e i π τ is given, then we use k = θ 2 2 ( 0 , q ) / θ 3 2 ( 0 , q ) , k = θ 4 2 ( 0 , q ) / θ 3 2 ( 0 , q ) , K = 1 2 π θ 3 2 ( 0 , q ) , and K = - i τ K , obtaining the values of the theta functions as in §20.14. …
20: 20.9 Relations to Other Functions
20.9.3 R F ( θ 2 2 ( z , q ) θ 2 2 ( 0 , q ) , θ 3 2 ( z , q ) θ 3 2 ( 0 , q ) , θ 4 2 ( z , q ) θ 4 2 ( 0 , q ) ) = θ 1 ( 0 , q ) θ 1 ( z , q ) z ,
20.9.4 R F ( 0 , θ 3 4 ( 0 , q ) , θ 4 4 ( 0 , q ) ) = 1 2 π ,
20.9.5 exp ( - π R F ( 0 , k 2 , 1 ) R F ( 0 , k 2 , 1 ) ) = q .