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1: 21.2 Definitions
§21.2(ii) Riemann Theta Functions with Characteristics
This function is referred to as a Riemann theta function with characteristics [ α β ] . …
21.2.7 θ [ 0 0 ] ( z | Ω ) = θ ( z | Ω ) .
Characteristics whose elements are either 0 or 1 2 are called half-period characteristics. For given Ω , there are 2 2 g g -dimensional Riemann theta functions with half-period characteristics. …
2: 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. …
3: 28.34 Methods of Computation
§28.34(i) Characteristic Exponents
§28.34(ii) Eigenvalues
  • (f)

    Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

  • 4: 28.7 Analytic Continuation of Eigenvalues
    §28.7 Analytic Continuation of Eigenvalues
    The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …
    28.7.4 n = 0 ( b 2 n + 2 ( q ) - ( 2 n + 2 ) 2 ) = 0 .
    5: 28.17 Stability as x ±
    The boundary of each region comprises the characteristic curves a = a n ( q ) and a = b n ( q ) ; compare Figure 28.2.1. …
    6: 21.3 Symmetry and Quasi-Periodicity
    §21.3(ii) Riemann Theta Functions with Characteristics
    21.3.4 θ [ α + m 1 β + m 2 ] ( z | Ω ) = e 2 π i α m 2 θ [ α β ] ( z | Ω ) .
    …For Riemann theta functions with half-period characteristics,
    21.3.6 θ [ α β ] ( - z | Ω ) = ( - 1 ) 4 α β θ [ α β ] ( z | Ω ) .
    7: 28.30 Expansions in Series of Eigenfunctions
    Let λ ^ m , m = 0 , 1 , 2 , , be the set of characteristic values (28.29.16) and (28.29.17), arranged in their natural order (see (28.29.18)), and let w m ( x ) , m = 0 , 1 , 2 , , be the eigenfunctions, that is, an orthonormal set of 2 π -periodic solutions; thus
    28.30.1 w m ′′ + ( λ ^ m + Q ( x ) ) w m = 0 ,
    8: 28.16 Asymptotic Expansions for Large q
    §28.16 Asymptotic Expansions for Large q
    9: 28.15 Expansions for Small q
    §28.15(i) Eigenvalues λ ν ( q )
    28.15.2 a - ν 2 - q 2 a - ( ν + 2 ) 2 - q 2 a - ( ν + 4 ) 2 - = q 2 a - ( ν - 2 ) 2 - q 2 a - ( ν - 4 ) 2 - .
    10: 21.1 Special Notation
    Uppercase boldface letters are g × g real or complex matrices. The main functions treated in this chapter are the Riemann theta functions θ ( z | Ω ) , and the Riemann theta functions with characteristics θ [ α β ] ( z | Ω ) . The function Θ ( ϕ | B ) = θ ( ϕ / ( 2 π i ) | B / ( 2 π i ) ) is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).