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21: 7.22 Methods of Computation

… ►Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions. …

22: 27.19 Methods of Computation: Factorization

… ►Trial division is one example. …

23: 2.1 Definitions and Elementary Properties

… ►

Examples

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Example

… ►For example, … ►For an example see (2.8.15). … ►For an example see §14.15(i). …

24: 9.17 Methods of Computation

… ►In the case of $\mathrm{Ai}\left(z\right)$, for example, this means that in the sectors we may integrate along outward rays from the origin with initial values obtained from §9.2(ii). … ►An example is provided by $\mathrm{Gi}\left(x\right)$ on the positive real axis. …

25: 27.6 Divisor Sums

… ►For example, … ►Examples include: …

26: 3.6 Linear Difference Equations

… ►An example is included in the next subsection. … ►

§3.6(vi) Examples

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Example 1. Bessel Functions

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Example 2. Weber Function

… ►For analyses and examples see Gautschi (1997b). …

27: 4.45 Methods of Computation

… ►As an example, take $x=9.47376$. … ►For example, $\arcsin x=\arctan \left(x{(1-x^2)}^{-1/2}\right)$. … ►For $x\in [-1/\mathrm{e},\mathrm{\infty })$ the principal branch $\mathrm{Wp}\left(x\right)$ can be computed by solving the defining equation $W{\mathrm{e}}^W=x$ numerically, for example, by Newton’s rule (§3.8(ii)). Initial approximations are obtainable, for example, from the power series (4.13.6) (with $t\ge 0$) when $x$ is close to $-1/\mathrm{e}$, from the asymptotic expansion (4.13.10) when $x$ is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of $x$. …

28: 10.42 Zeros

… ►For example, if $\nu$ is real, then the zeros of $I_{\nu }\left(z\right)$ are all complex unless for some positive integer $\mathrm{\ell }$, in which event $I_{\nu }\left(z\right)$ has two real zeros. …

29: 13.22 Zeros

… ►For example, if $\mu (\ge 0)$ is fixed and $\kappa (>0)$ is large, then the $r$th positive zero ${\varphi }_r$ of $M_{\kappa ,\mu }\left(z\right)$ is given by …

30: 19.15 Advantages of Symmetry

… ►For example, the computation of depolarization factors for solid ellipsoids is simplified considerably; compare (19.33.7) with Cronemeyer (1991). …