13.8. Our goal is to establish (13.60), which, with the aid of (13.59), can be put in the form eikr cos i47 (21+1) j(kr)Y,0(0) = (b) Use the fact that 1=0 (a) Explain why the expansion of the plane wave must be of the form eikr cos-c j(kr)Y,0(0) 1=0 12, 0) = (see Problem 9.18) to show (21)! (4) [1, Z) cijiler) = (211 / 42 v 1 {[10 (180 - } 20 cot 0. a ap eikr cos 8 d' d (cos 9) /(21)! [ as y, [(-1)'elle sin e- eikr cos 9 where the last step follows from the explicit form (9.142) for the raising operator in position space. (c) Use the explicit form (9.146) for Y(0, 0) to express this result in the form cji(kr) = (ikr)- 24/1 47 (21)! (21 + 1)! ds (Y/(0.4) Peikr cos 6 (d) Finally, isolate c, by evaluating this expression as r 0. Hint: For small r, ji(kr) behaves as (kr)/(21+ 1)!!, where (21+ 1)!!= (21 + 1) (21-1)... 5. 3.1.