Chester–Friedman–Ursell method

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11: 34.9 Graphical Method

§34.9 Graphical Method

►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …For an account of this method see Brink and Satchler (1993, Chapter VII). 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).

12: 17.18 Methods of Computation

§17.18 Methods of Computation

… ►Method (2) is very powerful when applicable (Andrews (1976, Chapter 5)); however, it is applicable only rarely. Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. ►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. Shanks (1955) applies such methods in several $q$-series problems; see Andrews et al. (1986).

13: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►General references for this subsection include Friedman (1990, pp. 4–6), Shilov (2013, pp. 249–256), Riesz and Sz.-Nagy (1990, Ch. 5, §82). … ►For $T$ to be actually self adjoint it is necessary to also show that $𝒟(T^{\ast })=𝒟(T)$, as it is often the case that $T$ and $T^{\ast }$ have different domains, see Friedman (1990, p 148) for a simple example of such differences involving the differential operator $\frac{d}{dx}$. … ►See Friedman (1990, pp. 233–252) for elementary discussions of both equations and the normalization process; and also the references in §1.18(ix). … ►The reader is referred to Coddington and Levinson (1955), Friedman (1990, Ch. 3), Titchmarsh (1962a), and Everitt (2005b, pp. 45–74) and Everitt (2005a, pp. 272–331), for detailed methods and results. … ►Friedman (1990) provides a useful introduction to both approaches; as does the conference proceeding Amrein et al. (2005), overviewing the combination of Sturm–Liouville theory and Hilbert space theory. …

14: 12.18 Methods of Computation

§12.18 Methods of Computation

►Because PCFs are special cases of confluent hypergeometric functions, the methods of computation described in §13.29 are applicable to PCFs. …

15: 36.13 Kelvin’s Ship-Wave Pattern

… ►The disturbance $z(\rho ,\varphi )$ can be approximated by the method of uniform asymptotic approximation for the case of two coalescing stationary points (36.12.11), using the fact that ${\theta }_{\pm }(\varphi )$ are real for and complex for $|\varphi |>{\varphi }_c$. … ►For further information see Lord Kelvin (1891, 1905) and Ursell (1960, 1994).

16: 14.26 Uniform Asymptotic Expansions

… ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.

17: Bibliography P

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Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§10.20(ii).

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Evaluation of Fermi-Dirac Integral. In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987), A. Cuyt (Ed.), Mathematics and Its Applications, Vol. 43, pp. 435–444.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§25.18(i).

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Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.

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ISSN 0006-3835, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(vii).

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A survey of numerical methods for the computation of Bessel function integrals. Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), pp. 249–265.

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Notes:

International conference on special functions: theory and computation (Turin, 1984)

External Links:

ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.

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External Links:

ISSN 0021-9991, Document, MathReview (J. G. Herriot), ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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18: Bibliography W

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Surface Waves. In Handbuch der Physik, Vol. 9, Part 3, pp. 446–778.

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External Links:

Online edition, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§11.12.

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Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.

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External Links:

Document, ZentralBlatt

Referenced by:

§33.5(ii).

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The method of Darboux. J. Approximation Theory 10 (2), pp. 159–171.

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External Links:

Document, MathReview (M. E. Noble), ZentralBlatt

Referenced by:

§2.10(iv).

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On a uniform treatment of Darboux’s method. Constr. Approx. 21 (2), pp. 225–255.

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External Links:

ISSN 0176-4276, Document, MathReview Entry, ZentralBlatt

Referenced by:

§2.10(iv).

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Asymptotic Approximations of Integrals. Academic Press Inc., Boston-New York.

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Notes:

Reprinted with corrections by SIAM, Philadelphia, PA, 2001.

External Links:

ISBN 0-12-762535-6, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§1.14(iv), §1.14(vii), §1.15(iv), §1.16(i), §1.16(ii), §1.16(iii), §1.16(iv), §1.16(v), §1.16(vi), §1.16(vii), §1.16(viii), §14.26, Ch.14, §2.10(iv), §2.3(i), §2.3(ii), §2.3(iii), §2.3(v), §2.4(i), §2.4(ii), §2.4(iv), §2.4(v), §2.4(vi), §2.5(iii), §2.5(i), §2.5(ii), §2.5(ii), §2.5(ii), §2.5(iii), §2.6(iii), §2.6(i), §2.6(i), §2.6(ii), §2.6(ii), §2.6(iii), §2.6(iv), §2.6(iv), Ch.2, §7.20(i), Ch.8, Ch.9, Profile Roderick S. C. Wong.

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19: 34.13 Methods of Computation

§34.13 Methods of Computation

►Methods of computation for $\mathit{3}j$ and $\mathit{6}j$ symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). ►For $\mathit{9}j$ symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). A review of methods of computation is given in Srinivasa Rao and Rajeswari (1993, Chapter VII, pp. 235–265). …

20: 19.4 Derivatives and Differential Equations

… ►An analogous differential equation of third order for $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ is given in Byrd and Friedman (1971, 118.03).