Solved: Find a second solution of Bessels equation of order one by computing the cn(r2)

Chapter 5, Problem 11

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Find a second solution of Bessels equation of order one by computing the cn(r2) and a of Eq. (24) of Section 5.6 according to the formulas (19) and (20) of that section. Some guidelines along the way of this calculation are the following. First, use Eq. (24) of this section to show that a1(1) and a 1(1) are 0. Then show that c1(1) = 0 and, from the recurrence relation, that cn(1) = 0 for n = 3, 5, .... Finally, use Eq. (25) to show that a2(r) = a0 (r + 1)(r + 3) , a4(r) = a0 (r + 1)(r + 3)(r + 3)(r + 5) , and that a2m(r) = (1)ma0 (r + 1)(r + 2m 1)(r + 3)(r + 2m + 1) , m 3. Then show that c2m(1) = (1) m+1 (Hm + Hm1)/22mm!(m 1)!, m 1