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11: 8.26 Tables

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Khamis (1965) tabulates $P(a,x)$ for $a=0.05(.05)10(.1)20(.25)70$, $0.0001\le x\le 250$ to 10D.

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Abramowitz and Stegun (1964, pp. 245–248) tabulates $E_n\left(x\right)$ for $n=2,3,4,10,20$, $x=0(.01)2$ to 7D; also $(x+n){\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2,3,4,10,20$, $x^{-1}=0(.01)0.1(.05)0.5$ to 6S.

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Pagurova (1961) tabulates $E_n\left(x\right)$ for $n=0(1)20$, $x=0(.01)2(.1)10$ to 4-9S; ${\mathrm{e}}^x{E}_n\left(x\right)$ for $n=2(1)10$, $x=10(.1)20$ to 7D; ${\mathrm{e}}^x{E}_p\left(x\right)$ for $p=0(.1)1$, $x=0.01(.01)7(.05)12(.1)20$ to 7S or 7D.

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Zhang and Jin (1996, Table 19.1) tabulates $E_n\left(x\right)$ for $n=1,2,3,5,10,15,20$, $x=0(.1)1,1.5,2,3,5,10,20,30,50,100$ to 7D or 8S.

12: 23 Weierstrass Elliptic and Modular
Functions

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13: Bibliography V

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Light Scattering by Small Particles. John Wiley and Sons. Inc., New York.

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External Links:

MathReview (P. M. Morse)

Referenced by:

§9.16.

Encodings:

BibTeX

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Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.

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External Links:

Document

Referenced by:

§14.31(i).

Encodings:

BibTeX

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

Encodings:

BibTeX

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14: 10.73 Physical Applications

… ►Laplace’s equation governs problems in heat conduction, in the distribution of potential in an electrostatic field, and in hydrodynamics in the irrotational motion of an incompressible fluid. … ►See Krivoshlykov (1994, Chapter 2, §2.2.10; Chapter 5, §5.2.2), Kapany and Burke (1972, Chapters 4–6; Chapter 7, §A.1), and Slater (1942, Chapter 4, §§20, 25). … ►In quantum mechanics the spherical Bessel functions arise in the solution of the Schrödinger wave equation for a particle in a central potential. …

15: 22.19 Physical Applications

… ►With appropriate scalings, Newton’s equation of motion for a pendulum with a mass in a gravitational field constrained to move in a vertical plane at a fixed distance from a fulcrum is … ►plays a prototypal role in classical mechanics (Lawden (1989, §5.2)), quantum mechanics (Schulman (1981, Chapter 29)), and quantum field theory (Pokorski (1987, p. 203), Parisi (1988, §14.6)). … ►We consider the case of a particle of mass 1, initially held at rest at displacement $a$ from the origin and then released at time $t=0$. … ►Such solutions include standing or stationary waves, periodic cnoidal waves, and single and multi-solitons occurring in diverse physical situations such as water waves, optical pulses, quantum fluids, and electrical impulses (Hasegawa (1989), Carr et al. (2000), Kivshar and Luther-Davies (1998), and Boyd (1998, Appendix D2.2)). …

16: 36 Integrals with Coalescing Saddles

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17: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

18: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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19: Bibliography W

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

BibTeX

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Introduction to Cyclotomic Fields. 2nd edition, Springer-Verlag, New York.

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External Links:

ISBN 0-387-94762-0, MathReview (T. Metsänkylä), ZentralBlatt

Referenced by:

§24.10(i), §24.10(ii), §24.16(ii), §24.17(iii).

Encodings:

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A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. 4th edition, Cambridge University Press, Cambridge.

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Notes:

Reprinted in 1988.

External Links:

ISBN 0-521-35883-3, MathReview, ZentralBlatt

Referenced by:

§22.19(i), §22.19(iv), §22.19(v), §23.21(i).

Encodings:

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Elliptic genera and quantum field theory. Comm. Math. Phys. 109 (4), pp. 525–536.

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External Links:

ISSN 0010-3616, Document, MathReview (J. Stasheff), ZentralBlatt

Referenced by:

1st item.

Encodings:

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20: 14.31 Other Applications

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§14.31(i) Toroidal Functions

►Applications of toroidal functions include expansion of vacuum magnetic fields in stellarators and tokamaks (van Milligen and López Fraguas (1994)), analytic solutions of Poisson’s equation in channel-like geometries (Hoyles et al. (1998)), and Dirichlet problems with toroidal symmetry (Gil et al. (2000)). … ►

§14.31(ii) Conical Functions

… ►These functions are also used in the Mehler–Fock integral transform (§14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). …