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21: 27.18 Methods of Computation: Primes

§27.18 Methods of Computation: Primes

►An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer $x$ is given in Crandall and Pomerance (2005, §3.7). … Oliveira e Silva has calculated $\pi \left(x\right)$ for $x={10}^{23}$, using the combinatorial methods of Lagarias et al. (1985) and Deléglise and Rivat (1996); see Oliveira e Silva (2006). …

22: 28.34 Methods of Computation

§28.34 Methods of Computation

… ►Methods available for computing the values of $w_{\text{I}}(\pi ;a,\pm q)$ needed in (28.2.16) include: … ►Methods for computing the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$, and ${\lambda }_{\nu }\left(q\right)$, defined in §§28.2(v) and 28.12(i), include: … ►

Methods described in §3.7(iv) applied to the differential equation (28.2.1) with the conditions (28.2.5) and (28.2.16).

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Solution of (28.2.1) by boundary-value methods; see §3.7(iii). This can be combined with §28.34(ii)(c).

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23: 29 Lamé Functions

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24: 5.21 Methods of Computation

§5.21 Methods of Computation

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25: 11.13 Methods of Computation

§11.13 Methods of Computation

… ►For a review of methods for the computation of ${𝐇}_{\nu }\left(z\right)$ see Zanovello (1975). … ►For complex variables the methods described in §§3.5(viii) and 3.5(ix) are available. … ►A comprehensive approach is to integrate the defining inhomogeneous differential equations (11.2.7) and (11.2.9) numerically, using methods described in §3.7. … ►In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …

26: 12.16 Mathematical Applications

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27: 31.18 Methods of Computation

§31.18 Methods of Computation

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28: Peter A. Clarkson

… ►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. … Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …

29: 33.23 Methods of Computation

§33.23 Methods of Computation

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§33.23(i) Methods for the Confluent Hypergeometric Functions

►The methods used for computing the Coulomb functions described below are similar to those in §13.29. … ►

§33.23(vi) Other Numerical Methods

►Curtis (1964a, §10) describes the use of series, radial integration, and other methods to generate the tables listed in §33.24. …

30: 24.19 Methods of Computation

§24.19 Methods of Computation

… ►For higher values of $n$ more efficient methods are available. …A similar method can be used for the Euler numbers based on (4.19.5). … ►Another method is based on the identities … ►We list here three methods, arranged in increasing order of efficiency. …