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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: 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. … Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. …
§21.2(ii) Riemann Theta Functions with Characteristics
§21.2(iii) Relation to Classical Theta Functions
3: 21 Multidimensional Theta Functions
Chapter 21 Multidimensional Theta Functions
4: 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
5: 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.

  • 6: 20.12 Mathematical Applications
    §20.12 Mathematical Applications
    §20.12(i) Number Theory
    §20.12(ii) Uniformization and Embedding of Complex Tori
    Thus theta functions “uniformize” the complex torus. …
    7: 21.8 Abelian Functions
    §21.8 Abelian Functions
    For every Abelian function, there is a positive integer n , such that the Abelian function can be expressed as a ratio of linear combinations of products with n factors of Riemann theta functions with characteristics that share a common period lattice. …
    8: 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 ) . …
    9: 20.11 Generalizations and Analogs
    §20.11 Generalizations and Analogs
    §20.11(ii) Ramanujan’s Theta Function and q -Series
    §20.11(iv) Theta Functions with Characteristics
    §20.11(v) Permutation Symmetry
    10: 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 .