Skip to main content
Log in

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

  • Published:
Journal of Geodesy Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • 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 

  • Balmino, G., On product of Legendre functions as encountered in geodynamics. Studia geoph. et geod., 22, 107–118, 1978.

    Google Scholar 

  • Burns, G., Introduction to group theory with applications, Academic Press, 1977.

  • Gerstl, M., On the recursive computation of the integrals of the associated Legendre functions, manuscripta geodaetica, 5, 181–199, 1980.

    Google Scholar 

  • 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 

  • Gradshteyn, I.S., and I. M. Ryzhik, Tables of integrals, series, and products, fifth ed., Academic Press, 1994.

  • Hagiwara, Y., Applications of Gaunt's integral to some problems in physical geodesy, J. Phy. Earth, 23, 311–321, 1975.

    Google Scholar 

  • Heiskanen, W.A. and H. Moritz, Physical geodesy, W.H. Freeman, New York, 1967.

    Google Scholar 

  • Hobson, E. W., The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Co., New York, second reprint, 1965.

    Google Scholar 

  • 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 

  • 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 

  • Paul, M.K., Recurrence relations for integrals of associated Legendre functions, Bulletin Géodésique, 52, 177–190, 1978.

    Google Scholar 

  • Wigner, E.P., Group theory and its application to the quantum mechanics of atomic spectra, Academic Press, 1959.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hwang, C. A method for computing the coefficients in the product-sum formula of associated Legendre functions. Journal of Geodesy 70, 110–116 (1995). https://doi.org/10.1007/BF00863422

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00863422

Keywords

Navigation