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31: 25.19 Tables
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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of for both real and complex . §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of , and §22.17 lists tables for some Dirichlet -functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.
32: 13.11 Series
33: 12.9 Asymptotic Expansions for Large Variable
34: 21.1 Special Notation
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positive integers. | |
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set of -dimensional vectors with elements in . | |
number of elements of the set . | |
set of all elements of the form “”. | |
set of all elements of , modulo elements of . Thus two elements of are equivalent if they are both in and their difference is in . (For an example see §20.12(ii).) | |
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35: 3.9 Acceleration of Convergence
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§3.9(ii) Euler’s Transformation of Series
… ►Euler’s transformation is usually applied to alternating series. … ►§3.9(iii) Aitken’s -Process
… ►§3.9(v) Levin’s and Weniger’s Transformations
►We give a special form of Levin’s transformation in which the sequence of partial sums is transformed into: …36: 25.11 Hurwitz Zeta Function
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has a meromorphic continuation in the -plane, its only singularity in being a simple pole at with residue .
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-Derivatives
►In (25.11.18)–(25.11.24) primes on denote derivatives with respect to . … ►where is Catalan’s constant: … ►As with fixed, , …37: 7.16 Generalized Error Functions
§7.16 Generalized Error Functions
►Generalizations of the error function and Dawson’s integral are and . …38: 17.5 Functions
39: 30.11 Radial Spheroidal Wave Functions
40: 12.11 Zeros
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►When the zeros are asymptotically given by and , where is a large positive integer and
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►Numerical calculations in this case show that corresponds to the th zero on the string; compare §7.13(ii).
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►For example, let the th real zeros of and , counted in descending order away from the point , be denoted by and , respectively.
…Here , denoting the th negative zero of the function (see §9.9(i)).
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►where , denoting the th negative zero of the function and
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