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1: 21.2 Definitions
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§21.2(ii) Riemann Theta Functions with Characteristics
โ–บThis function is referred to as a Riemann theta function with characteristics [ ๐œถ ๐œท ] . … โ–บ
21.2.7 ฮธ โข [ ๐ŸŽ ๐ŸŽ ] โก ( ๐ณ | ๐›€ ) = ฮธ โก ( ๐ณ | ๐›€ ) .
โ–บCharacteristics whose elements are either 0 or 1 2 are called half-period characteristics. For given ๐›€ , there are 2 2 โข g g -dimensional Riemann theta functions with half-period characteristics. …
2: 21.8 Abelian Functions
§21.8 Abelian Functions
โ–บFor 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
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§28.34(i) Characteristic Exponents
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§28.34(ii) Eigenvalues
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  • (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
    โ–บThe normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). … โ–บโ–บ
    28.7.4 n = 0 ( b 2 โข n + 2 โก ( q ) ( 2 โข n + 2 ) 2 ) = 0 .
    5: 28.17 Stability as x ±
    โ–บThe boundary of each region comprises the characteristic curves a = a n โก ( q ) and a = b n โก ( q ) ; compare Figure 28.2.1. …
    6: 21.3 Symmetry and Quasi-Periodicity
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    §21.3(ii) Riemann Theta Functions with Characteristics
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    21.3.4 ฮธ โข [ ๐œถ + ๐ฆ 1 ๐œท + ๐ฆ 2 ] โก ( ๐ณ | ๐›€ ) = e 2 โข ฯ€ โข i โข ๐œถ ๐ฆ 2 โข ฮธ โข [ ๐œถ ๐œท ] โก ( ๐ณ | ๐›€ ) .
    โ–บ …For Riemann theta functions with half-period characteristics, โ–บ
    21.3.6 ฮธ โข [ ๐œถ ๐œท ] โก ( ๐ณ | ๐›€ ) = ( 1 ) 4 โข ๐œถ ๐œท โข ฮธ โข [ ๐œถ ๐œท ] โก ( ๐ณ | ๐›€ ) .
    7: 28.30 Expansions in Series of Eigenfunctions
    โ–บLet ฮป ^ 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 โก ( x ) , m = 0 , 1 , 2 , , be the eigenfunctions, that is, an orthonormal set of 2 โข ฯ€ -periodic solutions; thus โ–บ
    28.30.1 w m ′′ + ( ฮป ^ m + Q โก ( x ) ) โข 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
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    §28.15(i) Eigenvalues ฮป ฮฝ โก ( q )
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    28.15.2 a ฮฝ 2 q 2 a ( ฮฝ + 2 ) 2 q 2 a ( ฮฝ + 4 ) 2 โ‹ฏ = q 2 a ( ฮฝ 2 ) 2 q 2 a ( ฮฝ 4 ) 2 โ‹ฏ .
    10: 21.1 Special Notation
    โ–บUppercase boldface letters are g × g real or complex matrices. โ–บThe main functions treated in this chapter are the Riemann theta functions ฮธ โก ( ๐ณ | ๐›€ ) , and the Riemann theta functions with characteristics ฮธ โข [ ๐œถ ๐œท ] โก ( ๐ณ | ๐›€ ) . โ–บThe function ฮ˜ โก ( ฯ• | ๐ ) = ฮธ โก ( ฯ• / ( 2 โข ฯ€ โข i ) | ๐ / ( 2 โข ฯ€ โข i ) ) is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).