Chapter 12: Q7-3P (page 562)
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Prove that the functions are orthogonal onwith respect to the weight function
Hint: Write the differential equationas, and see Sectionsand .
Substitute the P1(x)you found in Problems 4.3 or 5.3 into equation (10.6)to find, Plm(x); then let x=cos θto evaluate:
P32(cosθ)
Expand the following functions in Legendre series.
Show that the functions Plm(x)for each mare a set of orthogonal functions on (-1,1), that is, show that ∫-11Plm(x)Pnm(x)dx=0, l≠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.
Find the solutions of the following differential equations in terms of Bessel functions by using equations (16.1) and (16.2).
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