Chapter 12: Q7-6P (page 562)
Orthogonal functions are members of a function space, which is a bilinear vector space.
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Show that the functions Plm(x)for each mare a set of orthogonal functions on (-1,1), that is, show that โซ-11Plm(x)Pnm(x)dx=0, lโ n
Hint: Use the differential equations (10.1):
(1-x2) y"-2xy'+[l (l+1) -m2/1-x2] y=0 and follow the method of Section 7.
Expand the following functions in Legendre series.
Using (17.3) and (15.1) to (15.5), find the recursion relations for . In particular, show that .
To sketch the graph of for x from 0 to .
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 agreeing with the function for large x. If the small x approximation is not clear, plot it alone with the function over a small interval
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