344 Chapter 5 Higher Dimensions and Other Coordinates
Show that the coefficients are all zero afteranifμ^2 =n(n+ 1 ).
2.Derive the formula for the coefficientsbn, as shown in Eq. (10).
3.FindP 5 (x), first from the formulas for thea’s and second by using Eq. (9)
withn=4.
4.Verify Eqs. (6) and (7) forn= 0 , 1 ,2 and Eq. (9) forn= 2 ,3.
5.One of the solutions of( 1 −x^2 )y′′− 2 xy′=0isy(x)= 1 (μ^2 = 0 ).Find
another independent solution of this differential equation.
6.Show that the orthogonality relation for the eigenfunctions n(φ)=
Pn(cos(φ))is
∫π0n(φ)m(φ)sin(φ)dφ= 0 , n=m.7.Obtain the relationPn′+ 1 (x)=(n+ 1 )Pn(x)+xPn′(x)by differentiating Eq. (9) and eliminatingPn′− 1 between that and Eq. (8).
Note that Eq. (16) follows from this relation.
8.LetF=(x^2 − 1 )n. Show thatFsatisfies the differential equation
(
x^2 − 1)
F′= 2 nxF.9.Differentiate both sides of the preceding equationn+1 times to show that
thenth derivative ofFsatisfies Legendre’s equation (5). Use Leibniz’s rule
for derivatives of a product.
10.Obtain Eq. (6) by these manipulations:
a. Multiply through Eq. (9) byPn+ 1 ,integratefrom−1to1,andusethe
orthogonality ofPn+ 1 withPn− 1.
b. Replace( 2 n+ 1 )Pnby means of Eq. (8).
c.Pn+ 1 is orthogonal toxPn′− 1 , which is a polynomial of degreen.
d.Solve what remains for the desired integral.
11.Find the Legendre series for the functionf(x)=|x|,− 1 <x<1.
12.Find the Legendre series for the following function. Note thatf(x)− 1 / 2
is an odd function.f(x)={ 0 , − 1 <x<0,
1 , 0 <x<1.