# axially symmetric potential theory

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##### 1: 19.16 Definitions
###### §19.16(i) Symmetric Integrals
A fourth integral that is symmetric in only two variables is defined by … which is homogeneous and of degree $-a$ in the $z$’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\dots,n$. Thus $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ is symmetric in the variables $z_{j}$ and $z_{\ell}$ if the parameters $b_{j}$ and $b_{\ell}$ are equal. … …
##### 2: 19.18 Derivatives and Differential Equations
###### §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
###### §19.33(ii) Potential of a Charged Conducting Ellipsoid
The potential is
19.33.6 $1/C=R_{F}\left(a^{2},b^{2},c^{2}\right).$
##### 4: 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
• E. Bank and M. E. H. Ismail (1985) The attractive Coulomb potential polynomials. Constr. Approx. 1 (2), pp. 103–119.
• M. V. Berry and C. J. Howls (2010) Axial and focal-plane diffraction catastrophe integrals. J. Phys. A 43 (37), pp. 375206, 13.
• M. V. Berry (1966) Uniform approximation for potential scattering involving a rainbow. Proc. Phys. Soc. 89 (3), pp. 479–490.
• A. Bhattacharjie and E. C. G. Sudarshan (1962) A class of solvable potentials. Nuovo Cimento (10) 25, pp. 864–879.
• 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.
• ##### 6: 19.35 Other Applications
###### §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
###### 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
• 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.
• E. Neuman (2003) Bounds for symmetric elliptic integrals. J. Approx. Theory 122 (2), pp. 249–259.
• N. Nielsen (1906a) Handbuch der Theorie der Gammafunktion. B. G. Teubner, Leipzig (German).
• 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).
• Number Theory Web (website)
• ##### 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
• 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.
• 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.
• I. N. Sneddon (1966) Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.
• 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..
• R. P. Stanley (1989) Some combinatorial properties of Jack symmetric functions. Adv. Math. 77 (1), pp. 76–115.