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1: 20.2 Definitions and Periodic Properties
§20.2(i) Fourier Series
§20.2(ii) Periodicity and Quasi-Periodicity
The four points ( 0 , π , π + τ π , τ π ) are the vertices of the fundamental parallelogram in the z -plane; see Figure 20.2.1. …The theta functions are quasi-periodic on the lattice: …
§20.2(iv) z -Zeros
2: 21.2 Definitions
§21.2(i) Riemann Theta Functions
θ ( 𝐳 | 𝛀 ) is also referred to as a theta function with g components, a g -dimensional theta function or as a genus g theta function. For numerical purposes we use the scaled Riemann theta function θ ^ ( 𝐳 | 𝛀 ) , defined by (Deconinck et al. (2004)), …
§21.2(ii) Riemann Theta Functions with Characteristics
§21.2(iii) Relation to Classical Theta Functions
3: 20.4 Values at z = 0
§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 ,
Jacobi’s Identity
20.4.6 θ 1 ( 0 , q ) = θ 2 ( 0 , q ) θ 3 ( 0 , q ) θ 4 ( 0 , q ) .
20.4.12 θ 1 ′′′ ( 0 , q ) θ 1 ( 0 , q ) = θ 2 ′′ ( 0 , q ) θ 2 ( 0 , q ) + θ 3 ′′ ( 0 , q ) θ 3 ( 0 , q ) + θ 4 ′′ ( 0 , q ) θ 4 ( 0 , q ) .
4: 20.8 Watson’s Expansions
§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 .
5: 20.11 Generalizations and Analogs
§20.11 Generalizations and Analogs
For relatively prime integers m , n with n > 0 and m n even, the Gauss sum G ( m , n ) is defined by …
§20.11(ii) Ramanujan’s Theta Function and q -Series
§20.11(iv) Theta Functions with Characteristics
6: 21.3 Symmetry and Quasi-Periodicity
§21.3(i) Riemann Theta Functions
§21.3(ii) Riemann Theta Functions with Characteristics
…For Riemann theta functions with half-period characteristics, …
7: 20.7 Identities
§20.7(i) Sums of Squares
§20.7(ii) Addition Formulas
§20.7(v) Watson’s Identities
§20.7(vi) Landen Transformations
§20.7(vii) Derivatives of Ratios of Theta Functions
8: 20.1 Special Notation
Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21. 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 ) . This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951). …
9: 20.15 Tables
§20.15 Tables
Theta functions are tabulated in Jahnke and Emde (1945, p. 45). … Spenceley and Spenceley (1947) tabulates θ 1 ( x , q ) / θ 2 ( 0 , q ) , θ 2 ( x , q ) / θ 2 ( 0 , q ) , θ 3 ( x , q ) / θ 4 ( 0 , q ) , θ 4 ( x , q ) / θ 4 ( 0 , q ) to 12D for u = 0 ( 1 ) 90 , α = 0 ( 1 ) 89 , where u = 2 x / ( π θ 3 2 ( 0 , q ) ) and α is defined by (20.15.1), together with the corresponding values of θ 2 ( 0 , q ) and θ 4 ( 0 , q ) . Lawden (1989, pp. 270–279) tabulates θ j ( x , q ) , j = 1 , 2 , 3 , 4 , to 5D for x = 0 ( 1 ) 90 , q = 0.1 ( .1 ) 0.9 , and also q to 5D for k 2 = 0 ( .01 ) 1 . 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). …
10: 20.3 Graphics
§20.3(i) θ -Functions: Real Variable and Real Nome
See accompanying text
Figure 20.3.13: θ 4 ( π x , q ) , 0 x 2 , 0 q 0.99 . Magnify 3D Help
§20.3(ii) θ -Functions: Complex Variable and Real Nome
§20.3(iii) θ -Functions: Real Variable and Complex Lattice Parameter
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