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New identities for Legendre associated functions of integral order and degree
In the solution of the boundary value problems of mathematical physics in a separable 3-dimensional coordinate system, the shape of the boundary of the space may be such that the Green's function of the second order differential operator can be expanded as an infinite series of orthogonal functions....
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2016
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| _version_ | 1867613675898011648 |
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| access_status_str | Open Access |
| author | Schach, Stephen Ronald |
| author2 | Brundrit, Geoff |
| author_browse | Brundrit, Geoff Schach, Stephen Ronald |
| author_facet | Brundrit, Geoff Schach, Stephen Ronald |
| author_sort | Schach, Stephen Ronald |
| collection | Thesis |
| description | In the solution of the boundary value problems of mathematical physics in a separable 3-dimensional coordinate system, the shape of the boundary of the space may be such that the Green's function of the second order differential operator can be expanded as an infinite series of orthogonal functions. In many coordinate systems (such as the spherical, spheroidal and some cyclidal systems) these expansions are given in terms of Legendre associated functions of integral order and degree. Starting with Dougall's identities for Legendre associated functions of non-integral degree, new identities for infinite series of Legendre associated functions of integral degree are derived. Uniform convergence of each new identity is investigated in detail. The direct applicability of these identities is demonstrated by using them to verify theorems satisfied by the Dirichlet Green's function of the infinite half-space and of the interior of the prolate hemispheroid. The results and techniques are then generalized, and a sufficient condition found under which a generalized orthogonal function which satisfies Dougall's identity will also satisfy the new identity. This theorem is applied to the Legendre associated function, the generalized Legendre associated function and to the Jacobi function. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/16347 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:39:55.673Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/16347 New identities for Legendre associated functions of integral order and degree Schach, Stephen Ronald Brundrit, Geoff Mathematics and Applied Mathematics In the solution of the boundary value problems of mathematical physics in a separable 3-dimensional coordinate system, the shape of the boundary of the space may be such that the Green's function of the second order differential operator can be expanded as an infinite series of orthogonal functions. In many coordinate systems (such as the spherical, spheroidal and some cyclidal systems) these expansions are given in terms of Legendre associated functions of integral order and degree. Starting with Dougall's identities for Legendre associated functions of non-integral degree, new identities for infinite series of Legendre associated functions of integral degree are derived. Uniform convergence of each new identity is investigated in detail. The direct applicability of these identities is demonstrated by using them to verify theorems satisfied by the Dirichlet Green's function of the infinite half-space and of the interior of the prolate hemispheroid. The results and techniques are then generalized, and a sufficient condition found under which a generalized orthogonal function which satisfies Dougall's identity will also satisfy the new identity. This theorem is applied to the Legendre associated function, the generalized Legendre associated function and to the Jacobi function. 2016-01-12T11:19:25Z 2016-01-12T11:19:25Z 1973 Doctoral Thesis Doctoral PhD http://hdl.handle.net/11427/16347 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics and Applied Mathematics Schach, Stephen Ronald New identities for Legendre associated functions of integral order and degree |
| thesis_degree_str | Doctoral |
| title | New identities for Legendre associated functions of integral order and degree |
| title_full | New identities for Legendre associated functions of integral order and degree |
| title_fullStr | New identities for Legendre associated functions of integral order and degree |
| title_full_unstemmed | New identities for Legendre associated functions of integral order and degree |
| title_short | New identities for Legendre associated functions of integral order and degree |
| title_sort | new identities for legendre associated functions of integral order and degree |
| topic | Mathematics and Applied Mathematics |
| url | http://hdl.handle.net/11427/16347 |
| work_keys_str_mv | AT schachstephenronald newidentitiesforlegendreassociatedfunctionsofintegralorderanddegree |