Legendres differential equation reads where is some (non-negative) real number. (a) Assume a power series solution,
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Legendre’s differential equation reads
where ℓ is some (non-negative) real number.
(a) Assume a power series solution, and obtain a recursion relation for the constants
(b) Argue that unless the series truncates (which can only happen if ℓ is an integer), the solution will diverge at x = 1.
(c) When ℓ is an integer, the series for one of the two linearly independent solutions (either or depending on whether ℓ is even or odd) will truncate, and those solutions are called Legendre polynomials Pℓ(x).
Find P3(x), P3(x), P3(x), and P3(x) from the recursion relation. Leave your answer in terms of either a0 or a1.
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Related Book For
Introduction To Quantum Mechanics
ISBN: 9781107189638
3rd Edition
Authors: David J. Griffiths, Darrell F. Schroeter
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