axially symmetric potential theory

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2: 19.18 Derivatives and Differential Equations

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§19.18(i) Derivatives

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§19.18(ii) Differential Equations

… ►and two similar equations obtained by permuting

x, y, z

in (19.18.10). … ►The next four differential equations apply to the complete case of

R_{F}

and

R_{G}

in the form

R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};z_{1},z_{2}\right)

(see (19.16.20) and (19.16.23)). … ►Similarly, the function

u=R_{-a}\left(\tfrac{1}{2},\tfrac{1}{2};x+iy,x-iy\right)

satisfies an equation of axially symmetric potential theory: …

3: 19.33 Triaxial Ellipsoids

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19.33.1

S=3VR_{G}\left(a^{-2},b^{-2},c^{-2}\right),

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Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $V$ : volume and $S$ : area
Referenced by:: §19.33(i)
Permalink:: http://dlmf.nist.gov/19.33.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(i), §19.33 and Ch.19

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§19.33(ii) Potential of a Charged Conducting Ellipsoid

… ►The potential is ►

19.33.5

V(\lambda)=QR_{F}\left(a^{2}+\lambda,b^{2}+\lambda,c^{2}+\lambda\right),

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $Q$ : charge and $V(\lambda)$ : potential
Permalink:: http://dlmf.nist.gov/19.33.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(ii), §19.33 and Ch.19

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19.33.6

1/C=R_{F}\left(a^{2},b^{2},c^{2}\right).

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Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $C$ : electric capacity
Permalink:: http://dlmf.nist.gov/19.33.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(ii), §19.33 and Ch.19

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4: 19.15 Advantages of Symmetry

§19.15 Advantages of Symmetry

… ►The function

R_{-a}\left(b_{1},b_{2},\dots,b_{n};z_{1},z_{2},\dots,z_{n}\right)

(Carlson (1963)) reveals the full permutation symmetry that is partially hidden in

F_{D}

, and leads to symmetric standard integrals that simplify many aspects of theory, applications, and numerical computation. … ► ►Symmetry allows the expansion (19.19.7) in a series of elementary symmetric functions that gives high precision with relatively few terms and provides the most efficient method of computing the incomplete integral of the third kind (§19.36(i)). … ►

5: Bibliography B

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E. Bank and M. E. H. Ismail (1985) The attractive Coulomb potential polynomials. Constr. Approx. 1 (2), pp. 103–119.

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External Links:: ISSN 0176-4276, Document, Link, MathReview (W. A. Al-Salam)
Referenced by:: §18.39(iv), §18.39(iv).
Encodings:: BibTeX

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M. V. Berry and C. J. Howls (2010) Axial and focal-plane diffraction catastrophe integrals. J. Phys. A 43 (37), pp. 375206, 13.

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External Links:: ISSN 1751-8113, Document, FullText, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: (36.2.28), (36.2.29), §36.2(iv).
Encodings:: BibTeX

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M. V. Berry (1966) Uniform approximation for potential scattering involving a rainbow. Proc. Phys. Soc. 89 (3), pp. 479–490.

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External Links:: Document, ZentralBlatt
Referenced by:: §36.14(iii), §9.16.
Encodings:: BibTeX

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A. Bhattacharjie and E. C. G. Sudarshan (1962) A class of solvable potentials. Nuovo Cimento (10) 25, pp. 864–879.

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External Links:: Document, MathReview (S. Rosendorff), ZentralBlatt
Referenced by:: §15.18.
Encodings:: BibTeX

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J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow (2001) Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.

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External Links:: Document
Referenced by:: §29.19(i).
Encodings:: BibTeX

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6: 19.35 Other Applications

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§19.35(ii) Physical

►Elliptic integrals appear in lattice models of critical phenomena (Guttmann and Prellberg (1993)); theories of layered materials (Parkinson (1969)); fluid dynamics (Kida (1981)); string theory (Arutyunov and Staudacher (2004)); astrophysics (Dexter and Agol (2009)).

7: 18.38 Mathematical Applications

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Approximation Theory

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Complex Function Theory

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Zhedanov Algebra

►A symmetric Laurent polynomial is a function of the form … ►The solved Schrödinger equations of §18.39(i) involve shape invariant potentials, and thus are in the family of supersymmetric or SUSY potentials. …

8: Bibliography N

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J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

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External Links:: ISSN 0022-2488, Document, MathReview, ZentralBlatt
Referenced by:: §13.28(i).
Encodings:: BibTeX

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E. Neuman (2003) Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.

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External Links:: ISSN 0021-9045, Document, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §19.24(ii), §19.24(i), §19.9(i).
Encodings:: BibTeX

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N. Nielsen (1906a) Handbuch der Theorie der Gammafunktion. B. G. Teubner, Leipzig (German).

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Notes:: For a reprint with corrections see Nielsen (1965).
External Links:: ZentralBlatt (G. Kowalewski)
Referenced by:: §5.12, Ch.5, §6.10(i), §6.10(i), §7.9, §8.1.
Encodings:: BibTeX

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N. Nielsen (1965) Die Gammafunktion. Band I. Handbuch der Theorie der Gammafunktion. Band II. Theorie des Integrallogarithmus und verwandter Transzendenten. Chelsea Publishing Co., New York (German).

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Notes:: Both books are textually unaltered except for correction of errata.
External Links:: MathReview
Referenced by:: N. Nielsen (1906a), N. Nielsen (1906b).
Encodings:: BibTeX

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Number Theory Web (website)

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Notes:: References and links to software for factorization and primality testing.
External Links:: Home Page
Referenced by:: 6th item.
Encodings:: BibTeX

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9: 15.18 Physical Applications

… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). ►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).

10: Bibliography S

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B. E. Sagan (2001) The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. 2nd edition, Graduate Texts in Mathematics, Vol. 203, Springer-Verlag, New York.

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Notes:: First edition published by Wadsworth & Brooks/Cole Advanced Books & Software, 1991.
External Links:: ISBN 0-387-95067-2, MathReview, ZentralBlatt
Referenced by:: §26.19.
Encodings:: BibTeX

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B. Simon (1973) Resonances in

n

-body quantum systems with dilatation analytic potentials and the foundations of time-dependent perturbation theory. Ann. of Math. (2) 97, pp. 247–274.

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External Links:: ISSN 0003-486X, Document, Link, MathReview (K. Kumar)
Referenced by:: §1.18(viii).
Encodings:: BibTeX

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I. N. Sneddon (1966) Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.

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External Links:: MathReview (R. Shail), ZentralBlatt
Referenced by:: §10.22(iv).
Encodings:: BibTeX

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C. Snow (1952) Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..

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External Links:: MathReview, ZentralBlatt
Referenced by:: §13.23(iv), Ch.14, §31.3(iii).
Encodings:: BibTeX

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R. P. Stanley (1989) Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.

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External Links:: ISSN 0001-8708, Document, MathReview (Dennis White), ZentralBlatt
Referenced by:: §18.37(iii).
Encodings:: BibTeX

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