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Expansions in Series of Legendre Functions
Author(s) -
Love E. R.,
Hunter M. N.
Publication year - 1992
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-64.3.579
Subject(s) - legendre polynomials , mathematics , legendre function , laplace transform , series expansion , associated legendre polynomials , series (stratigraphy) , mathematical analysis , taylor series , function (biology) , pure mathematics , combinatorics , orthogonal polynomials , gegenbauer polynomials , classical orthogonal polynomials , paleontology , evolutionary biology , biology
The classical Laplace expansion of an ‘arbitrary’ function f in a series of Legendre polynomials is generalized, resulting in a (bilateral) series f ( x ) = ∑ n = ‐ ∞ ∞a n P v + n μ ( x )for −1 < x < 1, for suitable functions f and a range of real or complex parameters μ and ν in the Legendre functions. There is also a related expansion of zero, 0 = ∑ n = ‐ ∞ ∞b n P v + n μ ( x )for −1 < x < 1, where b n = (−1) n a n . Combining these gives f ( x ) = ∑ n = ‐ ∞ ∞c 2 nP v + 2 n μ ( x )for −1 < x < 1. The third of these expansions was suggested by a previously unpublished biorthogonality relation, but it is more easily approached via the first two expansions. A key role is played by Love's much earlier work on a generalized Neumann‐type integral [ 3 ].