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1: 14.30 Spherical and Spheroidal Harmonics
§14.30 Spherical and Spheroidal Harmonics
P n m ( x ) and Q n m ( x ) ( x > 1 ) are often referred to as the prolate spheroidal harmonics of the first and second kinds, respectively. P n m ( i x ) and Q n m ( i x ) ( x > 0 ) are known as oblate spheroidal harmonics of the first and second kinds, respectively. Segura and Gil (1999) introduced the scaled oblate spheroidal harmonics R n m ( x ) = e - i π n / 2 P n m ( i x ) and T n m ( x ) = i e i π n / 2 Q n m ( i x ) which are real when x > 0 and n = 0 , 1 , 2 , . …
2: Bibliography F
  • P. Falloon (2001) Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.
  • 3: Bibliography G
  • A. Gil and J. Segura (1998) A code to evaluate prolate and oblate spheroidal harmonics. Comput. Phys. Comm. 108 (2-3), pp. 267–278.
  • 4: Bibliography L
  • L.-W. Li, M. Leong, T.-S. Yeo, P.-S. Kooi, and K.-Y. Tan (1998a) Computations of spheroidal harmonics with complex arguments: A review with an algorithm. Phys. Rev. E 58 (5), pp. 6792–6806.
  • 5: Bibliography H
  • S. Hanish, R. V. Baier, A. L. Van Buren, and B. J. King (1970) Tables of Radial Spheroidal Wave Functions, Vols. 1-3, Prolate, m = 0 , 1 , 2 ; Vols. 4-6, Oblate, m = 0 , 1 , 2 . Technical report Naval Research Laboratory, Washington, D.C..
  • E. W. Hobson (1931) The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, London-New York.
  • L. K. Hua (1963) Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Translations of Mathematical Monographs, Vol. 6, American Mathematical Society, Providence, RI.
  • C. Hunter and B. Guerrieri (1982) The eigenvalues of the angular spheroidal wave equation. Stud. Appl. Math. 66 (3), pp. 217–240.
  • 6: Bibliography V
  • A. L. Van Buren, R. V. Baier, S. Hanish, and B. J. King (1972) Calculation of spheroidal wave functions. J. Acoust. Soc. Amer. 51, pp. 414–416.
  • A. L. Van Buren, R. V. Baier, and S. Hanish (1970) A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 6959 Naval Res. Lab.  Washingtion, D.C..
  • Van Buren (website) Mathieu and Spheroidal Wave Functions: Fortran Programs for their Accurate Calculation
  • B. Ph. van Milligen and A. López Fraguas (1994) Expansion of vacuum magnetic fields in toroidal harmonics. Comput. Phys. Comm. 81 (1-2), pp. 74–90.
  • H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.
  • 7: Bibliography
  • M. Abramowitz (1949) Asymptotic expansions of spheroidal wave functions. J. Math. Phys. Mass. Inst. Tech. 28, pp. 195–199.
  • J. C. Adams and P. N. Swarztrauber (1997) SPHEREPACK 2.0: A Model Development Facility. NCAR Technical Note Technical Report TN-436-STR, National Center for Atmospheric Research.
  • H. Alzer (1997a) A harmonic mean inequality for the gamma function. J. Comput. Appl. Math. 87 (2), pp. 195–198.
  • 8: Bibliography B
  • T. A. Beu and R. I. Câmpeanu (1983a) Prolate angular spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 187–192.
  • T. A. Beu and R. I. Câmpeanu (1983b) Prolate radial spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 177–185.
  • C. J. Bouwkamp (1947) On spheroidal wave functions of order zero. J. Math. Phys. Mass. Inst. Tech. 26, pp. 79–92.
  • W. J. Braithwaite (1973) Associated Legendre polynomials, ordinary and modified spherical harmonics. Comput. Phys. Comm. 5 (5), pp. 390–394.
  • T. Busch, B. Englert, K. Rzażewski, and M. Wilkens (1998) Two cold atoms in a harmonic trap. Found. Phys. 28 (4), pp. 549–559.