(Legendre functions of second kind)For the Legendre equation (7) on the interval −1 < x...

(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)

The recursion formula (16) holds for theQn’s as well as the Pn’s. Thus, withQo and Q1 in hand we can use (16) to obtainQ2, Q3, and so on. Do that: show that

and obtainQ3 (x)as well.