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Derivatives of addition theorems for Legendre functions

Published online by Cambridge University Press:  17 February 2009

D.E. Winch  and
P.H. Roberts
D.E. Winch
Affiliation:
Department of Applied Mathematics, University of Sydney, Sydney, 2006, Australia.
P.H. Roberts
Affiliation:
Institute of Geophysics and Planetary Physics, Los Angeles, California, 90024, U.S.A.
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Abstract

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Differentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 2 , October 1995 , pp. 212 - 234
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Chapman, S. and Bartels, J., Geomagnetism, (Oxford University Press, 1940) 621.Google Scholar
[2]Doodson, A. T., “The harmonic development of the tide generating potential”, Proc. R. Soc. Lond. A 100 (1922) 305–329.Google Scholar
[3]Elsasser, W. M., “A statistical analysis of the earth's internal magnetic field”, Phys. Rev. 60 (1941) 876–883.CrossRefGoogle Scholar
[4]Gegenbauer, L., “Das additions Theorem der Functionen Cvn(x)“, Wiener Sitzungsberichte 102 (1893) 942–950.Google Scholar
[5]James, R. W., “New tensor spherical harmonics, for application to the partial differential equations of mathematical physics”, Phil. Trans. R. Soc., Lond., A 281 (1976) 195–221.Google Scholar
[6]Roberts, P. H. and Winch, D. E., “On random rotations”, Adv. Appl. Prob. 16 (1984) 638–655.Google Scholar
[7]Whaler, K. A. and Gubbins, D., “Spherical harmonic analysis of the geomagnetic field: an example of a linear inverse problem”, Geophys. J. R. Astr. Soc. 65 (1981) 645–693.Google Scholar