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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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For Problems 1 to 4 , find one (simple) solution of each differential equation by series, and then find the second solution by the "reduction of order" method, Chapter 8 , Section 7 (e).
Multiply(5.8e)by and integrate from -1 to 1. To evaluate the middle term, integrate by parts.
Determine the raising and lowering operators for the spherical Bessel functions
.
The equation for the associated Legendre functions (and for Legendre functions when m=0) usually arises in the form (see, for example, Chapter 13, Section 7) 1/sinθ d/dθ (sinθ dy/dθ)+[l (l+1)-m2/sin2θ] y=0.
Make the change of variable x=cosθ, and obtain (10.1):
(1-x2) y"-2xy'+[l (l+1) -m2/1-x2] y=0
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