+0(x-5/2 ) +0(2-5/2)) ul(x) = a ( x1/2 = Problem 5.31. Show that solutions of the Bessel equation (4.59) have the asymptotics sin(x + b) (1/4 12) cos(x +b) u(x) = a x1/2 203/2 cos(x + b) (1/4 - v2) sin(x + b) + ) 2x3/2 (Hint: Show that after the transformation v(x) = Vzu(x) the Bessel equa- = x tion reads (cf. Problem 4.13) -V" (x) +9(x)(x) = 0, 1/4 - 12 9(x) = -1- x2 Now use a modified Prfer transform with h(x) = -9(x) (set p(x) = 1, 2 V r(x) = 0) and verify ,(x) = 1+ 1/4 - 12 +0(x-3), Po(x) 0(x-3), 2x2 v (2) as x +00.) = = - = zau" + zu' + (22 v2)u = 0, = VEC. (4.59) +0(x-5/2 ) +0(2-5/2)) ul(x) = a ( x1/2 = Problem 5.31. Show that solutions of the Bessel equation (4.59) have the asymptotics sin(x + b) (1/4 12) cos(x +b) u(x) = a x1/2 203/2 cos(x + b) (1/4 - v2) sin(x + b) + ) 2x3/2 (Hint: Show that after the transformation v(x) = Vzu(x) the Bessel equa- = x tion reads (cf. Problem 4.13) -V" (x) +9(x)(x) = 0, 1/4 - 12 9(x) = -1- x2 Now use a modified Prfer transform with h(x) = -9(x) (set p(x) = 1, 2 V r(x) = 0) and verify ,(x) = 1+ 1/4 - 12 +0(x-3), Po(x) 0(x-3), 2x2 v (2) as x +00.) = = - = zau" + zu' + (22 v2)u = 0, = VEC. (4.59)