Riemann%20%CE%BE-function
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1: 21.7 Riemann Surfaces
§21.7 Riemann Surfaces
►§21.7(i) Connection of Riemann Theta Functions to Riemann Surfaces
►In almost all applications, a Riemann theta function is associated with a compact Riemann surface. … ►is a Riemann matrix and it is used to define the corresponding Riemann theta function. … ►§21.7(iii) Frobenius’ Identity
…2: 25.1 Special Notation
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►The main function treated in this chapter is the Riemann zeta function
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This notation was introduced in Riemann (1859).
►The main related functions are the Hurwitz zeta function
, the dilogarithm , the polylogarithm (also known as Jonquière’s function
), Lerch’s transcendent , and the Dirichlet -functions
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nonnegative integers. | |
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3: 21.2 Definitions
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§21.2(i) Riemann Theta Functions
… ►For numerical purposes we use the scaled Riemann theta function , defined by (Deconinck et al. (2004)), …Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. … ►§21.2(ii) Riemann Theta Functions with Characteristics
… ►It is a translation of the Riemann theta function (21.2.1), multiplied by an exponential factor: …4: 25.20 Approximations
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Cody et al. (1971) gives rational approximations for in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are , , , . Precision is varied, with a maximum of 20S.
Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of and , , for (23D).
5: 25.3 Graphics
6: 25.6 Integer Arguments
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§25.6(i) Function Values
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25.6.3
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25.6.4
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§25.6(ii) Derivative Values
… ►§25.6(iii) Recursion Formulas
…7: 21.10 Methods of Computation
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§21.10(i) General Riemann Theta Functions
… ►§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
… ►Tretkoff and Tretkoff (1984). Here a Hurwitz system is chosen to represent the Riemann surface.
Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.