(Legendre functions of second kind)For the Legendre equation (7) on the interval −1 <x < 1, we obtained the bounded solutiony(x) = Pn(x).In this exercise we seek a second LI solution, denoted asQn(x)and called theLegendre function of the second kind.Then the general solution of (7) can be expressed as
y (x) = APn (x) +BQn(x).(11.1)
For the special casen =0, solve (7) and show that a second LI solution is In [(1 + x)/1 − x)]. Scaling this solution by 1/2, we define
Sketch the graph ofQ0(x)on −1 <x <1, and notice that |Q0(x)| → ∞ asx→ ±1.
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