Abel–Plana formula
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21: 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.
22: 29.20 Methods of Computation
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►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6.
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►§29.15(i) includes formulas for normalizing the eigenvectors.
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23: 14.28 Sums
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§14.28(ii) Heine’s Formula
…24: Need Help?
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25: 5.21 Methods of Computation
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►For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3).
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26: 12.16 Mathematical Applications
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►PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs.
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27: 30.10 Series and Integrals
28: 31.18 Methods of Computation
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►Subsequently, the coefficients in the necessary connection formulas can be calculated numerically by matching the values of solutions and their derivatives at suitably chosen values of ; see Laĭ (1994) and Lay et al. (1998).
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29: 25.10 Zeros
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§25.10(ii) Riemann–Siegel Formula
… ►Sign changes of are determined by multiplying (25.9.3) by to obtain the Riemann–Siegel formula: ►
25.10.3
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►Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of in the critical strip are on the critical line (van de Lune et al. (1986)).
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