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1: 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 .
2: 22.2 Definitions
k = θ 2 2 ( 0 , q ) θ 3 2 ( 0 , q ) ,
k = θ 4 2 ( 0 , q ) θ 3 2 ( 0 , q ) ,
22.2.4 sn ( z , k ) = θ 3 ( 0 , q ) θ 2 ( 0 , q ) θ 1 ( ζ , q ) θ 4 ( ζ , q ) = 1 ns ( z , k ) ,
22.2.7 sd ( z , k ) = θ 3 2 ( 0 , q ) θ 2 ( 0 , q ) θ 4 ( 0 , q ) θ 1 ( ζ , q ) θ 3 ( ζ , q ) = 1 ds ( z , k ) ,
22.2.11 p q ( z , k ) = θ p ( z | τ ) / θ q ( z | τ ) ,
3: 22.16 Related Functions
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 ) . …
4: 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. …
5: 20.4 Values at z = 0
20.4.1 θ 1 ( 0 , q ) = θ 2 ( 0 , q ) = θ 3 ( 0 , q ) = θ 4 ( 0 , q ) = 0 ,
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 ) .
6: 22.11 Fourier and Hyperbolic Series
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 ,
22.11.2 cn ( z , k ) = 2 π K k n = 0 q n + 1 2 cos ( ( 2 n + 1 ) ζ ) 1 + q 2 n + 1 ,
22.11.3 dn ( z , k ) = π 2 K + 2 π K n = 1 q n cos ( 2 n ζ ) 1 + q 2 n .
22.11.4 cd ( z , k ) = 2 π K k n = 0 ( - 1 ) n q n + 1 2 cos ( ( 2 n + 1 ) ζ ) 1 - q 2 n + 1 ,
7: 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 ) ,
20.7.3 θ 2 2 ( 0 , q ) θ 4 2 ( z , q ) = θ 3 2 ( 0 , q ) θ 1 2 ( z , q ) + θ 4 2 ( 0 , q ) θ 2 2 ( z , q ) ,
20.7.4 θ 2 2 ( 0 , q ) θ 3 2 ( z , q ) = θ 4 2 ( 0 , q ) θ 1 2 ( z , q ) + θ 3 2 ( 0 , q ) θ 2 2 ( z , q ) .
20.7.5 θ 3 4 ( 0 , q ) = θ 2 4 ( 0 , q ) + θ 4 4 ( 0 , q ) .
8: 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 .
9: 20.3 Graphics
§20.3(i) θ -Functions: Real Variable and Real Nome
See accompanying text
Figure 20.3.2: θ 1 ( π x , q ) , 0 x 2 , q = 0. … Magnify
See accompanying text
Figure 20.3.3: θ 2 ( π x , q ) , 0 x 2 , q = 0. … Magnify
§20.3(ii) θ -Functions: Complex Variable and Real Nome
See accompanying text
Figure 20.3.18: θ 1 ( 0.1 | u + i v ) , - 1 u 1 , 0.005 v 0.5 . …1 of z is chosen arbitrarily since θ 1 vanishes identically when z = 0 . Magnify 3D Help
10: 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 ) .