New Relations Among Associated Legendre Functions and Spherical Harmonics
Several new relations among associated Legendre functions (ALFs) are derived, most of which relate a product of an ALF with trigonometric functions to a weighted summation over ALFs, where the weights only depend on the degree and order of the ALF. These relations are, for example, useful in applica...
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| Format: | Journal Article |
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Springer - Verlag
2005
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| Online Access: | http://hdl.handle.net/20.500.11937/19446 |
| _version_ | 1848750036274380800 |
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| author | Claessens, Sten |
| author_facet | Claessens, Sten |
| author_sort | Claessens, Sten |
| building | Curtin Institutional Repository |
| collection | Online Access |
| description | Several new relations among associated Legendre functions (ALFs) are derived, most of which relate a product of an ALF with trigonometric functions to a weighted summation over ALFs, where the weights only depend on the degree and order of the ALF. These relations are, for example, useful in applications such as the computation of geopotential coefficients and computation of ellipsoidal corrections in geoid modelling. The main relations are presented in both their unnormalised and fully normalised ($4\pi$-normalised) form. Several approaches to compute the weights involved are discussed, and it is shown that the relations can also be applied in the case of first- and second-order derivatives of ALFs, which may be of use in analysis of satellite gradiometry data. Finally, the derived relations are combined to provide new identities among ALFs, which contain no dependency on the colatitudinal coordinate other than that in the ALFs themselves. |
| first_indexed | 2025-11-14T07:30:27Z |
| format | Journal Article |
| id | curtin-20.500.11937-19446 |
| institution | Curtin University Malaysia |
| institution_category | Local University |
| last_indexed | 2025-11-14T07:30:27Z |
| publishDate | 2005 |
| publisher | Springer - Verlag |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | curtin-20.500.11937-194462018-10-02T04:31:44Z New Relations Among Associated Legendre Functions and Spherical Harmonics Claessens, Sten Spherical Harmonics Associated Legendre Functions Several new relations among associated Legendre functions (ALFs) are derived, most of which relate a product of an ALF with trigonometric functions to a weighted summation over ALFs, where the weights only depend on the degree and order of the ALF. These relations are, for example, useful in applications such as the computation of geopotential coefficients and computation of ellipsoidal corrections in geoid modelling. The main relations are presented in both their unnormalised and fully normalised ($4\pi$-normalised) form. Several approaches to compute the weights involved are discussed, and it is shown that the relations can also be applied in the case of first- and second-order derivatives of ALFs, which may be of use in analysis of satellite gradiometry data. Finally, the derived relations are combined to provide new identities among ALFs, which contain no dependency on the colatitudinal coordinate other than that in the ALFs themselves. 2005 Journal Article http://hdl.handle.net/20.500.11937/19446 10.1007/s00190-005-0483-9 Springer - Verlag fulltext |
| spellingShingle | Spherical Harmonics Associated Legendre Functions Claessens, Sten New Relations Among Associated Legendre Functions and Spherical Harmonics |
| title | New Relations Among Associated Legendre Functions and Spherical Harmonics |
| title_full | New Relations Among Associated Legendre Functions and Spherical Harmonics |
| title_fullStr | New Relations Among Associated Legendre Functions and Spherical Harmonics |
| title_full_unstemmed | New Relations Among Associated Legendre Functions and Spherical Harmonics |
| title_short | New Relations Among Associated Legendre Functions and Spherical Harmonics |
| title_sort | new relations among associated legendre functions and spherical harmonics |
| topic | Spherical Harmonics Associated Legendre Functions |
| url | http://hdl.handle.net/20.500.11937/19446 |