Relationships between solid spherical and toroidal harmonics

Preprint English OPEN
Majic, Matt ; Ru, Eric C. Le (2018)
  • Subject: Physics - Classical Physics | Mathematical Physics

We derive new relationships expressing solid spherical harmonics as series of toroidal harmonics and vice versa. The expansions include regular and irregular spherical harmonics, ring and axial toroidal harmonics of even and odd parity about the plane of the torus. The expansion coefficients are given in terms of a recurrence relation. As an example application we apply one of the expansions to express the potential of a charged conducting torus on a basis of spherical harmonics.
  • References (14)
    14 references, page 1 of 2

    1 r a 2 1 r a 2

    [4] Panayiotis Vafeas, Polycarpos K. Papadopoulos, Ping-Ping Ding, and Dominique Lesselier. Mathematical and numerical analysis of low-frequency scattering from a PEC ring torus in a conductive medium. Applied Mathematical Modelling, 40(13- 14):6477{6500, 2016. ISSN 0307904X. doi:10.1016/j.apm.2016.01.053. URL pii/S0307904X1630049X.

    [5] B. P. Kondratyev, A. S. Dubrovskii, and N. G. Trubitsina. Potential in the Clearance of torus: Development on spherical functions. Technical Physics, 57(12):1613{1617, 2012. ISSN 1063-7842. doi:10.1134/S1063784212120134. URL http: //

    [6] Marcello Cavallaro Jr, Mohamed A. Gharbi, Daniel A. Beller, Simon Copar, Zheng Shi, Randall D. Kamien, Shu Yang, Tobias Baumgart, and Kathleen J. Stebe. Ring around the colloid. Soft Matter, 9(38):9099, 2013. ISSN 1744-683X. doi:10.1039/c3sm51167g. URL

    [7] Mark Andrews. Alternative separation of Laplace's equation in toroidal coordinates and its application to electrostatics. Journal of Electrostatics, 64(10):664{672, 2006. ISSN 03043886. doi:10.1016/j.elstat.2005.11.005.

    [8] Philip M. Morse and Herman Feshbach. Methods of Theoretical Physics [Part 1 Chaps 1-8] 1953.pdf, 1953.

    [9] Howard S. Cohl and Joel E. Tohline. A Compact Cylindrical Green's Function Expansion for the Solution of Potential Problems. The Astrophysical Journal, 527(1):86{101, 1999. ISSN 0004-637X. doi:10.1086/308062. URL http://stacks.

    [10] Sameir M Ali Hamed. Response to 'Comments on 'Exact eld expressions for circular loop antennas using spherical functions expansion". IEEE Transactions on Antennas and Propagation, 62(8):4434{4435, 2014. ISSN 0018926X. doi: 10.1109/TAP.2014.2329002.

    [11] NN Lebedev. Special functions and their applications. revised english edition. Translated and edited by Richard A. Silverman, 1965.

    [12] Georg Jansen. Transformation properties of spheroidal multipole moments and potentials. Journal of Physics A: Mathematical and General, 33(7):1375, 2000.

  • Metrics
    No metrics available
Share - Bookmark