theta%0Afunctions
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1: 20.2 Definitions and Periodic Properties
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§20.2(i) Fourier Series
… ►§20.2(ii) Periodicity and Quasi-Periodicity
… ►The four points are the vertices of the fundamental parallelogram in the -plane; see Figure 20.2.1. …The theta functions are quasi-periodic on the lattice: … ►§20.2(iv) -Zeros
…2: 21.2 Definitions
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§21.2(i) Riemann Theta Functions
… ► is also referred to as a theta function with components, a -dimensional theta function or as a genus theta function. ►For numerical purposes we use the scaled Riemann theta function , 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 = 0
4: 20.8 Watson’s Expansions
5: 20.11 Generalizations and Analogs
§20.11 Generalizations and Analogs
… ►For relatively prime integers with and even, the Gauss sum is defined by … ►§20.11(ii) Ramanujan’s Theta Function and -Series
… ► … ►§20.11(iv) Theta Functions with Characteristics
…6: 21.3 Symmetry and Quasi-Periodicity
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§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
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§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
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►Sometimes the theta functions are called the Jacobian or classical theta functions to distinguish them from generalizations; compare Chapter 21.
►Primes on the symbols indicate derivatives with respect to the argument of the function.
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►Jacobi’s original notation: , , , , respectively, for , , , , where .
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►Neville’s notation: , , , , respectively, for , , , , where again .
This notation simplifies the relationship of the theta functions to Jacobian elliptic functions (§22.2); see Neville (1951).
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