Journal of Geodesy

, Volume 70, Issue 1–2, pp 110–116 | Cite as

A method for computing the coefficients in the product-sum formula of associated Legendre functions

  • Cheinway Hwang


The product of two associated Legendre functions can be represented by a finite series in associated Legendre functions with unique coefficients. In this study a method is proposed to compute the coefficients in this product-sum formula. The method is of recursive nature and is based on the straightforward polynomial form of the associated Legendre function's factor. The method is verified through the computation of integrals of products of two associated Legendre functions over a given interval and the computation of integrals of products of two Legendre polynomials over [0,1]. These coefficients are basically constant and can be used in any future related applications. A table containing the coefficients up to degree 5 is given for ready reference.


Legendre Polynomial Legendre Function Polynomial Form Related Application Ready Reference 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Banerji, S., Note on the product of any number of Legendre functions of different degrees, Bull. Calcutta Math. Soc., 11 (1919–20), 179–185, 1920.Google Scholar
  2. Balmino, G., On product of Legendre functions as encountered in geodynamics. Studia geoph. et geod., 22, 107–118, 1978.Google Scholar
  3. Burns, G., Introduction to group theory with applications, Academic Press, 1977.Google Scholar
  4. Gerstl, M., On the recursive computation of the integrals of the associated Legendre functions, manuscripta geodaetica, 5, 181–199, 1980.Google Scholar
  5. Giacaglia, G.E.D., Transformations of spherical harmonics and applications to geodesy and satellite theory, Studia geoph. et geod., 24, 1–11, 1980.Google Scholar
  6. Gradshteyn, I.S., and I. M. Ryzhik, Tables of integrals, series, and products, fifth ed., Academic Press, 1994.Google Scholar
  7. Hagiwara, Y., Applications of Gaunt's integral to some problems in physical geodesy, J. Phy. Earth, 23, 311–321, 1975.Google Scholar
  8. Heiskanen, W.A. and H. Moritz, Physical geodesy, W.H. Freeman, New York, 1967.Google Scholar
  9. Hobson, E. W., The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Co., New York, second reprint, 1965.Google Scholar
  10. Hwang, C., Spectral analysis using orthonormal functions with a case study on the sea surface topography, Geophysical Journal International, 115, 1148–1160, 1993.Google Scholar
  11. Mainville, A., The altimetery-gravimetry problem using orthonormal base functions, Dept. of Geodetic Science and Surveying, Rep. No. 373, The Ohio State University, Columbus, 1987.Google Scholar
  12. Paul, M.K., Recurrence relations for integrals of associated Legendre functions, Bulletin Géodésique, 52, 177–190, 1978.Google Scholar
  13. Wigner, E.P., Group theory and its application to the quantum mechanics of atomic spectra, Academic Press, 1959.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Cheinway Hwang
    • 1
  1. 1.Department of Civil EngineeringNational Chiao Tung UniversityHsinchuTaiwan

Personalised recommendations