Abstract
A new method is presented for Fourier decomposition of the Helmholtz Green function in cylindrical coordinates, which is equivalent to obtaining the solution of the Helmholtz equation for a general ring source. The Fourier coefficients of the Green function are split into their half advanced + half retarded and half advanced–half retarded components, and closed form solutions for these components are then obtained in terms of a Horn function and a Kampé de Fériet function respectively. Series solutions for the Fourier coefficients are given in terms of associated Legendre functions, Bessel and Hankel functions and a hypergeometric function. These series are derived either from the closed form 2-dimensional hypergeometric solutions or from an integral representation, or from both. A simple closed form far-field solution for the general Fourier coefficient is derived from the Hankel series. Numerical calculations comparing different methods of calculating the Fourier coefficients are presented. Fourth order ordinary differential equations for the Fourier coefficients are also given and discussed briefly.
Similar content being viewed by others
References
Mathews J., Walker R.L.: Mathematical Methods of Physics. 2nd edn. Addison Wesley, New York (1973)
Overfelt P.L.: Near fields of the constant current thin circular loop antenna of arbitrary radius. IEEE Trans. Antennas Propagat. 44, 166–171 (1996)
Werner D.H.: An exact integration procedure for vector potentials of thin circular loop antennas. IEEE Trans. Antennas Propagat. 44, 157–165 (1996)
Conway J.T.: New exact solution procedure for the near fields of the thin circular loop antenna. IEEE Trans. Antennas Propagat. 53, 509–517 (2005)
Prentice P.R.: The acoustic ring source and its application to propeller acoustics. Proc. R. Soc. Lond. A 437, 629–644 (1992)
Matviyenko G.: On the azimuthal Fourier components of the Green’s function for the Helmholtz equation in three dimensions. J. Math. Phys. 36, 5159–5169 (1995)
Gradshteyn I.S., Rhyzik I.M.: Table of Integrals, Series and Products, 7th edn. Academic, New York (2007)
Cohl H.S., Tohline J.E.: A compact cylindrical Green’s function expansion for the solution of potential problems. Astrophys. J. 527, 86–101 (1999)
Watson G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944)
Morse P.M., Feshbach H.L: Methods of Theoretical Physics, vol. 1. McGraw-Hill, New York (1953)
Erdélyi A. et al.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)
Srivastava H.M., Karlsson P.W.: Multiple Gaussian Hypergeometric Series. Ellis Horwood, Chichester (1985)
Conway, J.T., Cohl, H.S.: Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function (2009). arXiv:0910.1193v1 [math-ph]
Conway J.T.: Fourier series for elliptic integrals and some generalizations via hypergeometric series. Intgr. Transf. Spec. F. 19, 305–315 (2008)
Wolfram S.: The Mathematica Book, 5th edn. Wolfram Media, Champaign (2003)
Prudnikov A.P., Brychkov Yu.A., Marichev O.I.: Integrals and Series, vol. 3, More Special Functions. Gordon and Breach, New York (1990)
Abramowitz M., Stegun I.S.: Handbook of Mathematical Functions. Dover, New York (1972)
Cohl H.S., Tohline J.E., Rau A.R.P., Srivastava H.M.: Developments in determining the gravitational potential using toroidal functions. Astron. Nachr. 321(5/6), 363–372 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Conway, J.T., Cohl, H.S. Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function. Z. Angew. Math. Phys. 61, 425–443 (2010). https://doi.org/10.1007/s00033-009-0039-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00033-009-0039-6