Find a second solution of Bessels equation of order one by

Chapter 5, Problem 11

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Find a second solution of Bessels equation of order one by computing the cn(r2) and aof Eq. (24) of Section 5.6 according to the formulas (19) and (20) of that section. Someguidelines along the way of this calculation are the following. First, use Eq. (24) of thissection to show that a1(1) and a1(1) are 0. Then show that c1(1) = 0 and, from therecurrence relation, that cn(1) = 0 for n = 3, 5, .... Finally, use Eq. (25) to show thata2(r) = a0(r + 1)(r + 3), a4(r) = a0(r + 1)(r + 3)(r + 3)(r + 5)and thata2m(r) = (1)ma0(r + 1)(r + 2m 1)(r + 3)(r + 2m + 1), m 3.Then show thatc2m(1) = (1)m+1(Hm + Hm1)/22mm!(m 1)!, m 1.