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41: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►Examples of applications to noninteger and/or complex variables are as follows. ►

Scattering at complex energies. See for example McDonald and Nuttall (1969).

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Regge poles at complex values of $\mathrm{\ell }$. See for example Takemasa et al. (1979).

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Gravitational radiation. See for example Berti and Cardoso (2006).

►For further examples see Humblet (1984). …

42: 33.23 Methods of Computation

… ►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. … ►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). …

43: 1.10 Functions of a Complex Variable

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Examples

… ►For examples of applications see Olver (1997b, pp. 252–254). … ►

Example

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44: 10.74 Methods of Computation

… ►For $z\in ℂ$ the function $H_{\nu }^{(1)}\left(z\right)$, for example, can always be computed in a stable manner in the sector $0\le \mathrm{ph}z\le \pi$ by integrating along rays towards the origin. … ►Suppose, for example, $\nu =n\in 0,1,2,\mathrm{\dots }$, and $x\in (0,\mathrm{\infty })$. … ►Necessary values of the first derivatives of the functions are obtained by the use of (10.6.2), for example. …

45: 13.29 Methods of Computation

… ►In the following two examples Olver’s algorithm (§3.6(v)) can be used. ►

Example 1

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Example 2

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46: 19.14 Reduction of General Elliptic Integrals

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§19.14(i) Examples

… ►If no such branch point is accessible from the interval of integration (for example, if the integrand is ${(t(3-t)(4-t))}^{-3/2}$ and the interval is [1,2]), then no method using this assumption succeeds. …

47: 26.9 Integer Partitions: Restricted Number and Part Size

… ►An example is provided in Figure 26.9.1. … ►The conjugate to the example in Figure 26.9.1 is $6+5+4+2+1+1+1$. … ►Equations (26.9.2)–(26.9.3) are examples of closed forms that can be computed explicitly for any positive integer $k$. …

48: 32.9 Other Elementary Solutions

… ►For examples and plots see Clarkson (2003a) and Milne et al. (1997). … ►For examples and plots see Clarkson (2005). … ►For further examples of algebraic solutions see Andreev and Kitaev (2002), Boalch (2005, 2006), Gromak et al. (2002, §48), Hitchin (2003), Masuda (2003), and Mazzocco (2001b). …

49: 26.13 Permutations: Cycle Notation

… ►For the example (26.13.2), this decomposition is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(1,3\right)\left(2,3\right)\left(2,5\right)\left(5,7\right)\left(6,8\right).$ … ►Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by $\left(1,3,2,5,7\right)\left(6,8\right)=\left(2,3\right)\left(1,2\right)\left(4,5\right)\left(3,4\right)\left(2,3\right)\left(3,4\right)\left(4,5\right)\left(6,7\right)\left(5,6\right)\left(7,8\right)\left(6,7\right)$: $inv(\left(1,3,2,5,7\right)\left(6,8\right))=11$.

50: 2.10 Sums and Sequences

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Example

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