Homework Set # 9 - Due November 4

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      1) 9.7.1 (Notice that since the solution we are asked to consider depends on r rather than r there will no bet angular dependence which means that n=0). 

     2)   Solve Hermite's equation using a series solution (as we did in class for Legendre equation). The equation is given by:

                                                     y''-2xy'+2 alpha y=0.     (1)

            a) Propose a series solution of the form 7.28 (see book) and find the values of k for which the series solves (1).
            b) For each value of k found in (a) obtain an expression for a_{j+2} in terms of a_j and provide the first three terms in the series for y(x) (you should obtain a series in even powers of x and another series in odd powers of x).
            c) Now consider that alpha is 0 or a positive integer. For alpha=0, 1, 2, and 3 find which of the 2 solutions corresponds to a polynomial (finite series) and provide the corresponding expression. These are the Hermite's polynomials. Compare your results with the polynomials shown in Table 18.1 (in the book) and find the value of a_0 in each case.

     3) 15.3.1
     
     4) 15.3.2

     5)  15.2.3 (Hint: P_2n(0)=(-1)ⁿ(2n)!/2²ⁿ/(n!)² and P_2n+1(0)=0 for n=0, 1, 2, ...)

     6)  15.2.12

     7)  15.2.13  (Hint: use expression 15.79)