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Chapter 12: Q8-1P (page 562)
Find the norm of each of the following functions on the given interval and state the normalized function.
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Show that
Write (10.7) with m replaced by -m; then use Problem 7 to show that Pl-m(x) = (-1)m (l-m)!/(l+m)! Plm(x).
Comment: This shows that (10.7) is a solution of (10.1) when m is negative.
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|
Expand the following functions in Legendre series.
Solve the differential equations in Problems 5 to 10 by the Frobenius method; observe that you get only one solution. (Note, also, that the two values of are equal or differ by an integer, and in the latter case the larger gives the one solution.) Show that the conditions of Fuchs's theorem are satisfied. Knowing that the second solution is x times the so
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