Riemann%20zeta%20function
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1: 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.
2: 25.3 Graphics
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3: 26.12 Plane Partitions
4: 25.5 Integral Representations
§25.5 Integral Representations
►§25.5(i) In Terms of Elementary Functions
… ►§25.5(ii) In Terms of Other Functions
… ►For similar representations involving other theta functions see Erdélyi et al. (1954a, p. 339). … ►§25.5(iii) Contour Integrals
…5: 25.6 Integer Arguments
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25.6.3
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6: Bibliography B
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A program for computing the Riemann zeta function for complex argument.
Comput. Phys. Comm. 20 (3), pp. 441–445.
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7: Bibliography K
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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8: 25.12 Polylogarithms
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►The notation was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828):
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►The special case is the Riemann zeta function: .
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►Further properties include
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►In terms of polylogarithms
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9: 20.10 Integrals
§20.10 Integrals
►§20.10(i) Mellin Transforms with respect to the Lattice Parameter
… ►Here again denotes the Riemann zeta function (§25.2). … ►§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
… ►For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193). …10: Software Index
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►‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support.
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►In the list below we identify four main sources of software for computing special functions.
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Commercial Software.
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►The following are web-based software repositories with significant holdings in the area of special functions.
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Open Source | With Book | Commercial | |||||||||||||||||||||||
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20 Theta Functions | |||||||||||||||||||||||||
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Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.