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Chapter 12: Q8-5P (page 562)
Find the norm of each of the following functions on the given interval and state the normalized function
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Find the best (in the least squares sense) second-degree polynomial approximation to each of the given functions over the interval -1<x<1.
|x|
To study the approximations in the table, a computer plot on the same axes the given function together with its small x approximation and its asymptotic approximation. Use an interval large enough to show the asymptotic approximation agrees with the function for large x . If the small x approximation is not clear, plot it alone with the function over a small interval .
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 ,.
Verify equations (10.3) and (10.4).
(10.3) : (1-x2) u"-2 (m+1) xu'+[l(l+1) - m(m+1)] u=0
(10.4) : (1-x2) (u')" -2 [(m+1)+1] x(u')'+ [l(l+1) - (m+1)(m+2)]u'=0
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.
x-x3
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