ps7Xtra - e.g p.209 of our text(a show that this is a...

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MATH 405: Problem Set 7 Additional Problems due Monday 11/6/2006 1. In Kreyszig Section 5.8 the expansion of a function f in terms of a Legendre series f ( x ) = Σ a n P n ( x ) is discussed, where it is given that: a n = 2 n + 1 2 Z 1 - 1 f ( x ) P n ( x ) dx Derive this relation explicitly (without trying to get the constant in front, which comes from the norm of P n ). Note that there is no weighting function for the Legendre polynomials when they are functions of x , i.e. R ( x ) = 1. 2. Consider the following ODE for y ( x ): (1 - x 2 ) y 00 - xy 0 + ky = 0 This equation is close to but not the same as Legendre’s equation, and is known as the Chebyshev equation. Its solutions are known as the Chebyshev polynomials T ( x ) (see
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Unformatted text preview: e.g. p.209 of our text). (a) show that this is a Sturm-Liouville equation, and find the orthogonality relation between the Chebyshev polynomials. (b) following what we did in class to solve Legendre’s equation, obtain the recursion relation for a power series solution to the Chebyshev equation and show that it truncates with the choice k = n 2 (where n is an integer), leading to the Chebyshev polynomials T n ( x ). 3. Derivative of the factorial? Using the definition of the gamma function Γ( n ), take the derivative d Γ /dn . Can you write the answer in terms of other gamma functions?...
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This note was uploaded on 07/23/2008 for the course MATH 405 taught by Professor Belmonte during the Fall '06 term at Penn State.

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