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Chapter 12: Q8-6P (page 562)
Multiply(5.8e)by and integrate from -1 to 1. To evaluate the middle term, integrate by parts.
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Show that dl-m/dxl-m (x2-1)l=(l-m)!/(l+m)! (x2-1)m dl+m/dxl+m (x2-1)l.
Hint: Write(x2-1)l=(x-1)l(x+1)land find the derivatives by Leibniz' rule.
Verify the formula stated for and in terms of and and also find the value of the, and .
Verify the formula stated for and in terms of and and also find the value of the (x) and .
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/sin胃 d/d胃 (sin胃 dy/d胃)+[l (l+1)-m2/sin2胃] y=0.
Make the change of variable x=cos胃, and obtain (10.1):
(1-x2) y"-2xy'+[l (l+1) -m2/1-x2] y=0
Prove that the functions are orthogonal onwith respect to the weight function
Hint: Write the differential equationas, and see Sectionsand .
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