Chapter 12: Q 9-2P (page 562)
Expand the following functions in Legendre series.
Short Answer

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Chapter 12: Q 9-2P (page 562)
Expand the following functions in Legendre series.

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Expand each of the following polynomials in a Legendre series. You should get the same results that you got by a different method in the corresponding problems in Section 5.
3x2+x-1
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 that the differential equation in Problemis not Fuchsian. Solve it by separation of variables to find the obvious solutionconst. and a second solution in the form of an integral. Show that the second solution is not expandable in a Frobenius series.
Use the Section 15 recursion relations and (17.4) to obtain the following recursion relations for spherical Bessel functions. We have written them for, but they are valid forand for the ,.
To show that .
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