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11: 27.15 Chinese Remainder Theorem

… ►Most of the calculation can be done with five-digit integers as follows. Choose four relatively prime moduli $m_1,m_2,m_3$, and $m_4$ of five digits each, for example $2^{16}-3$, $2^{16}-1$, $2^{16}+1$, and $2^{16}+3$. …By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod $m_1$), (mod $m_2$), (mod $m_3$), and (mod $m_4$), respectively. Because each residue has no more than five digits, the arithmetic can be performed efficiently on these residues with respect to each of the moduli, yielding answers $a_1(mod{m}_1)$, $a_2(mod{m}_2)$, $a_3(mod{m}_3)$, and $a_4(mod{m}_4)$, where each $a_j$ has no more than five digits. … ►Details of a machine program describing the method together with typical numerical results can be found in Newman (1967). …

12: 33 Coulomb Functions

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13: 36.15 Methods of Computation

… ►Close to the origin $𝐱=\mathrm{𝟎}$ of parameter space, the series in §36.8 can be used. … ►Close to the bifurcation set but far from $𝐱=\mathrm{𝟎}$, the uniform asymptotic approximations of §36.12 can be used. … ►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real $t$-axis containing all real critical points of $\mathrm{\Phi }$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\exp \left(\mathrm{i}\mathrm{\Phi }\right)$. … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …

14: 34.9 Graphical Method

§34.9 Graphical Method

… ►Thus, any analytic expression in the theory, for example equations (34.3.16), (34.4.1), (34.5.15), and (34.7.3), may be represented by a diagram; conversely, any diagram represents an analytic equation. …For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

15: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. …Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21). … ► (16) $3{\kappa }^2+2$ should be $3{\kappa }^{2}c+2$). …

16: Browsers

… ►MathML has long been considered to be important for representing mathematics on the web; it provides for much better accessibility, reusability and scalability to different displays. Although we are not yet encoding our graphics in SVG, will be looking to make that conversion in the future. …

17: 9.17 Methods of Computation

… ►For large $|z|$ the asymptotic expansions of §§9.7 and 9.12(viii) should be used instead. …However, in the case of $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$ this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). … ►As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation. … ►In these cases boundary-value methods need to be used instead; see §3.7(iii). … ►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. …

18: 26.20 Physical Applications

… ►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). … ►Other applications to problems in engineering, crystallography, biology, and computer science can be found in Beckenbach (1981) and Graham et al. (1995).

19: 6.18 Methods of Computation

… ►The attainable accuracy of the asymptotic expansions can be increased considerably by exponential improvement. Also, other ranges of $\mathrm{ph}z$ can be covered by use of the continuation formulas of §6.4. … ►For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998). … ►Lastly, the continued fraction (6.9.1) can be used if $|z|$ is bounded away from the origin. … ►Power series, asymptotic expansions, and quadrature can also be used to compute the functions $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$. …

20: 10.74 Methods of Computation

… ►The power-series expansions given in §§10.2 and 10.8, together with the connection formulas of §10.4, can be used to compute the Bessel and Hankel functions when the argument $x$ or $z$ is sufficiently small in absolute value. … ►Furthermore, the attainable accuracy can be increased substantially by use of the exponentially-improved expansions given in §10.17(v), even more so by application of the hyperasymptotic expansions to be found in the references in that subsection. ►For large positive real values of $\nu$ the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used. … ►And since there are no error terms they could, in theory, be used for all values of $z$; however, there may be severe cancellation when $|z|$ is not large compared with $n^2$. … ►Then $J_n\left(x\right)$ and $Y_n\left(x\right)$ can be generated by either forward or backward recurrence on $n$ when , but if $n>x$ then to maintain stability $J_n\left(x\right)$ has to be generated by backward recurrence on $n$, and $Y_n\left(x\right)$ has to be generated by forward recurrence on $n$. …