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31: 25.18 Methods of Computation

… ►For recent investigations see, for example, van de Lune et al. (1986) and Odlyzko (1987). …

32: 26.17 The Twelvefold Way

… ►See also Example 3 in §26.18. …

33: 27.16 Cryptography

… ►For example, a code maker chooses two large primes $p$ and $q$ of about 400 decimal digits each. …

34: 27.21 Tables

… ►8 gives examples of primitive roots of all primes $\le 9973$; Table 24. …

35: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

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Example

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36: 29.20 Methods of Computation

… ►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). …

37: 34.8 Approximations for Large Parameters

… ►For example, …

38: 3.8 Nonlinear Equations

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Example

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Example

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Example

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Example. Wilkinson’s Polynomial

… ►Consider, for example, (3.8.9). …

39: 15.19 Methods of Computation

… ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. … ►For example, in the half-plane $\mathrm{\Re }z\le \frac{1}{2}$ we can use (15.12.2) or (15.12.3) to compute $F(a,b;c+N+1;z)$ and $F(a,b;c+N;z)$, where $N$ is a large positive integer, and then apply (15.5.18) in the backward direction. …

40: 21.1 Special Notation

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$g,h$	positive integers.
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$S_1/S_2$	set of all elements of $S_1$, modulo elements of $S_2$. Thus two elements of $S_1/S_2$ are equivalent if they are both in $S_1$ and their difference is in $S_2$. (For an example see §20.12(ii).)
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… ►The function $\mathrm{\Theta }(\mathit{\varphi }|𝐁)=\theta \left(\mathit{\varphi }/(2\pi \mathrm{i})\right|𝐁/(2\pi \mathrm{i}))$ is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).