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Jacobi theta functions

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1: 20.11 Generalizations and Analogs
For m = 1 , 2 , 3 , 4 , n = 1 , 2 , 3 , 4 , and m n , define twelve combined theta functions φ m , n ( z , q ) by
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 .
2: 20.2 Definitions and Periodic Properties
§20.2(i) Fourier Series
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 ) .
3: 22.2 Definitions
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 ) ,
22.2.5 cn ( z , k ) = θ 4 ( 0 , q ) θ 2 ( 0 , q ) θ 2 ( ζ , q ) θ 4 ( ζ , q ) = 1 nc ( z , k ) ,
22.2.6 dn ( z , k ) = θ 4 ( 0 , q ) θ 3 ( 0 , q ) θ 3 ( ζ , q ) θ 4 ( ζ , q ) = 1 nd ( 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 ) ,
22.2.9 sc ( z , k ) = θ 3 ( 0 , q ) θ 4 ( 0 , q ) θ 1 ( ζ , q ) θ 2 ( ζ , q ) = 1 cs ( z , k ) .
4: 23.15 Definitions
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 .
5: 20.15 Tables
20.15.1 sin α = θ 2 2 ( 0 , q ) / θ 3 2 ( 0 , q ) = k .
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). …
6: 20.1 Special Notation
The main functions treated in this chapter are the theta functions θ j ( z | τ ) = θ j ( z , q ) where j = 1 , 2 , 3 , 4 and q = e i π τ . … Primes on the θ symbols indicate derivatives with respect to the argument of the θ function. … Jacobi’s original notation: Θ ( z | τ ) , Θ 1 ( z | τ ) , H ( z | τ ) , H 1 ( z | τ ) , respectively, for θ 4 ( u | τ ) , θ 3 ( u | τ ) , θ 1 ( u | τ ) , θ 2 ( u | τ ) , where u = z / θ 3 2 ( 0 | τ ) . … 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 ) . … McKean and Moll’s notation: ϑ j ( z | τ ) = θ j ( π z | τ ) , j = 1 , 2 , 3 , 4 . …
7: 20.10 Integrals
20.10.1 0 x s - 1 θ 2 ( 0 | i x 2 ) d x = 2 s ( 1 - 2 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
20.10.2 0 x s - 1 ( θ 3 ( 0 | i x 2 ) - 1 ) d x = π - s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 1 ,
20.10.3 0 x s - 1 ( 1 - θ 4 ( 0 | i x 2 ) ) d x = ( 1 - 2 1 - s ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) , s > 0 .
20.10.4 0 e - s t θ 1 ( β π 2 | i π t 2 ) d t = 0 e - s t θ 2 ( ( 1 + β ) π 2 | i π t 2 ) d t = - s sinh ( β s ) sech ( s ) ,
20.10.5 0 e - s t θ 3 ( ( 1 + β ) π 2 | i π t 2 ) d t = 0 e - s t θ 4 ( β π 2 | i π t 2 ) d t = s cosh ( β s ) csch ( s ) .
8: 20.13 Physical Applications
The functions θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , provide periodic solutions of the partial differential equation …
20.13.4 π 4 α t n = - e - ( n π + z ) 2 / ( 4 α t ) = θ 3 ( z | i 4 α t / π ) ,
20.13.5 π 4 α t n = - ( - 1 ) n e - ( n π + z ) 2 / ( 4 α t ) = θ 4 ( z | i 4 α t / π ) .
In the singular limit τ 0 + , the functions θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …
9: 20.4 Values at z = 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 .
Jacobi’s Identity
20.4.6 θ 1 ( 0 , q ) = θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 4 ( 0 , q ) .
10: 20.9 Relations to Other Functions
20.9.1 k = θ 2 2 ( 0 | τ ) / θ 3 2 ( 0 | τ )
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 ,
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). …