Journal of Algorithms and Optimization                  
Journal of Algorithms and Optimization(JAO)
Accurate Positions of Branch Points of Minimum Magnitudes and Their Associated Spheroidal Eigenvalues
Full Paper(PDF, 144KB)
The Newton- Raphson method with two complex variable is utilized to compute the branch points with minimum magnitudes and their associated eigenvalues in the first quadrant of the complex plane with the parameter c =kF, where k is a complex wave number, and F is the semifocal length of the spheroidal system. The efficient numerical method which is applied to the spheroidal eigenvalue equation and the equation of its partial derivative with respect to the eigenvalue, is used to simultaneously solve the two complex variables, the branch point and its associated eigenvalue with a high precision, from which they can be tabulated for references.
Keywords:Branch Points; Spheroidal Eigenvalues; Newton-Raphson’s Method
Author: Tam Do-Nhat1
1.7-116 Candlewood Cr. Waterloo City, Ontario Province, N2L5M8, Canada
  1. T. Do-Nhat, “Accurate power series for eigenvalues of spheroidal angle functions and their convergence radii,” Can. J. Phys., vol. 89, pp. 1083-1099, 2011.
  2. J. Meixner, F.W. Schafke, and G. Wolf, Mathieu functions and spheroidal functions and their mathematical foundations, New York: Springer- Verlag, pp. 102-110, 1980.
  3. C. Hunter and B. Guerrieri, “The eigenvalues of the angular spheroidal wave equation,” Stud. Appl. Math., vol. 66, pp. 217-240, 1982.
  4. B.E. Barrowes, K. O'Neill, T.M. Grzegorczyk, and J.A. Kong, “On the asymptotic expansion of the spheroidal wave function and its eigenvalues for complex size parameter,” Stud. Appl. Math., vol. 113, pp. 271-301, 2004.
  5. T. Oguchi, “Eigenvalues of spheroidal wave functions and their branch points for complex values of propagation constants,” Radio. Sci., vol. 5, pp. 1207-1214, Aug. 1970.
  6. T. Barakat, K. Abodayeh and O. Al-Dossary, “The asymptotic iteration method for the angular spheroidal eigenvalues with arbitrary complex size parameter c,” Can. J. Phys., vol. 84, pp. 121-129, 2006.
  7. S. Skokhodov and D. Khristoforov, “Calculation of the branch points of the eigenfunctions corresponding to wave spheroidal functions,” Computation Mathematics and Mathematical Physics, vol. 46, pp. 1132-1146, 2006.
  8. L. Li, M. Leong, T. Yeo, P. Kooi, and K. Tan, “Computations of spheroidal harmonics with complex arguments: A review with an algorithm,” Phys. Rev. E, vol. 58, pp. 6792-6806, Nov. 1998.
  9. L. Li, X. Kang, and M. Leong, Spheroidal wave functions in electromagnetic theory, Wiley, New York, pp. 13-26, 2001.
  10. C. Flammer, Spheroidal wave functions, Stanford, California: Stanford University Press, pp. 16-29, 1957.
  11. M. Abramowitz and A. Stegun, Handbook of mathematical functions, Dover, New York, pp. 751-759, 1970.
  12. J. Stratton, P. Morse, L. Chu, J. Little, and F. Corbato, Spheroidal wave functions, John Wiley and Sons, New York, 1956.
  13. Morse and Feshbach, Methods of theoretical physics (Part 2), McGraw-Hill, Boston, pp. 642-644, 1953.
  14. T. Do-Nhat, “Asymptotic expansion of the Mathieu and prolate spheroidal eigenvalues for large parameter c,” Can. J. Phys., vol. 77, pp. 635-652, 1999.
  15. T. Do-Nhat, “Asymptotic expansions of the oblate spheroidal eigenvalues and wave functions for large parameter c,” Can. J. Phys., vol. 79, pp. 813-831, 2001.