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1: 20.2 Definitions and Periodic Properties
§20.2(i) Fourier Series
§20.2(ii) Periodicity and Quasi-Periodicity
The theta functions are quasi-periodic on the lattice: …
§20.2(iii) Translation of the Argument by Half-Periods
§20.2(iv) z -Zeros
2: 20.11 Generalizations and Analogs
§20.11 Generalizations and Analogs
§20.11(iv) Theta Functions with Characteristics
§20.11(v) Permutation Symmetry
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 …
3: 27.13 Functions
Jacobi (1829) notes that r 2 ( n ) is the coefficient of x n in the square of the theta function ϑ ( x ) :
27.13.4 ϑ ( x ) = 1 + 2 m = 1 x m 2 , | x | < 1 .
(In §20.2(i), ϑ ( x ) is denoted by θ 3 ( 0 , x ) .) …
27.13.6 ( ϑ ( x ) ) 2 = 1 + 4 n = 1 ( δ 1 ( n ) - δ 3 ( n ) ) x n ,
Mordell (1917) notes that r k ( n ) is the coefficient of x n in the power-series expansion of the k th power of the series for ϑ ( x ) . …
4: 9.8 Modulus and Phase
9.8.1 Ai ( x ) = M ( x ) sin θ ( x ) ,
9.8.2 Bi ( x ) = M ( x ) cos θ ( x ) ,
9.8.4 θ ( x ) = arctan ( Ai ( x ) / Bi ( x ) ) .
(These definitions of θ ( x ) and ϕ ( x ) differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).) … As x increases from - to 0 each of the functions M ( x ) , M ( x ) , | x | - 1 / 4 N ( x ) , M ( x ) N ( x ) , θ ( x ) , ϕ ( x ) is increasing, and each of the functions | x | 1 / 4 M ( x ) , θ ( x ) , ϕ ( x ) is decreasing. …
5: 21.2 Definitions
§21.2(i) Riemann Theta Functions
21.2.1 θ ( z | Ω ) = n g e 2 π i ( 1 2 n Ω n + n z ) .
21.2.2 θ ^ ( z | Ω ) = e - π [ z ] [ Ω ] - 1 [ z ] θ ( z | Ω ) .
§21.2(ii) Riemann Theta Functions with Characteristics
21.2.5 θ [ α β ] ( z | Ω ) = n g e 2 π i ( 1 2 [ n + α ] Ω [ n + α ] + [ n + α ] [ z + β ] ) .
6: 10.18 Modulus and Phase Functions
10.18.1 M ν ( x ) e i θ ν ( x ) = H ν ( 1 ) ( x ) ,
where M ν ( x ) ( > 0 ) , N ν ( x ) ( > 0 ) , θ ν ( x ) , and ϕ ν ( x ) are continuous real functions of ν and x , with the branches of θ ν ( x ) and ϕ ν ( x ) fixed by …
10.18.9 N ν 2 ( x ) = M ν 2 ( x ) + M ν 2 ( x ) θ ν 2 ( x ) = M ν 2 ( x ) + 4 ( π x M ν ( x ) ) 2 ,
10.18.11 tan ( ϕ ν ( x ) - θ ν ( x ) ) = M ν ( x ) θ ν ( x ) M ν ( x ) = 2 π x M ν ( x ) M ν ( x ) ,
7: 33.2 Definitions and Basic Properties
33.2.8 H ± ( η , ρ ) = e ± i θ ( η , ρ ) ( 2 i ρ ) + 1 ± i η U ( + 1 ± i η , 2 + 2 , 2 i ρ ) ,
33.2.9 θ ( η , ρ ) = ρ - η ln ( 2 ρ ) - 1 2 π + σ ( η ) ,
8: 21 Multidimensional Theta Functions
Chapter 21 Multidimensional Theta Functions
9: 21.9 Integrable Equations
§21.9 Integrable Equations
Typical examples of such equations are the Korteweg–de Vries equation …
See accompanying text
Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3). … Magnify
10: 21.10 Methods of Computation
§21.10(i) General Riemann Theta Functions
§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
  • Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.