1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

4.4 • POWER SERIES FUNCTIONS 153

and termwise differentiation shows that its derivative is

We leave as an exercise to show that the radius of convergence of these series is
infinity. The Bessel function J 1 ( z) of order 1 is kn own to sat isfy the differential

equation J 1 (z) = - J 0 (z).

-------... EXERCISES FOR SECTION 4.4

1. Prove part (iii) of Theorem 4. 16.
   00 OOTl OOn.


2. Consider the series E zn, E ~.and E ~.
   n::O n=l n=l

(a) Show that each series has radius of convergence 1.

(b) Show that the first series converges nowhere on C 1 (0) = {z: lzl = l}.

(c) Show that the second series converges everywhere on C 1 (0).

(d) It turns out th~t the third series converges everywhere on C1 (0), except

at t he point z = 1. This is not easy to prove. Give it a try.

3. Find the radius of convergence of the following.
   00
   (a) g (z) = E (-1)" (;:)!'
   n=O
   00
   (b) h(z) = E n!zn.
   n=O

( ) Cf ( Z ) = L,,, ~ ( 2n4,,2 +l - 3n+4 6n2 )n Z n ·

n =O

(d) ( ) ~ (n!)

(^2) n
g Z = L,,, (2n)! Z.
n=O
00
(e) h(z) = E (2- (-1)")" z".
n = O
(g) g (z) = n~O ( !:t; r z".