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Chapter 12: Q7-6P (page 562)
Orthogonal functions are members of a function space, which is a bilinear vector space.
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Verify by direct substitution that the text solution of equation (16.3) and your solutions in the problems above are correct. Also prove in general that the solution (16.2) given for (16.1) is correct. Hint: These are exercises in partial differentiation. To verify the solution (16.4) of (16.3), we would change variables from x,y to say z, u where , and show that if x,y satisfy then u , z satisfy, .
Expand the following functions in Legendre series.
f(x) = P'n (x).
Hint: For I鈮� n, 鈭�-11 P'n(x)Pl(x) dx=0 (Why?); for l<n, integrate by parts.
Find the best (in the least squares sense) second-degree polynomial approximation to each of the given functions over the interval -1<x<1.
肠辞蝉蟺虫
Expand each of the following polynomials in a Legendre series. You should get the same results that you got by a different method in the corresponding problems in Section 5.
3x2+x-1
Question: Use the Section 15 recursion relations and (17.4) to obtain the following recursion relations for spherical Bessel functions. We have written them for, but they are valid forand for the
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