1 5 2 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES
y-+-------------- x
Figure 4.4 Choosing 8 to prove that J' (zo) = g (zo).
00• EXAMPLE 4. 24 Show that E (n + 1) zn = (J~z) 2 for all z E D 1 (O).
n=O
00Solution We know from Theorem 4.1;2 that f (z) = 1 ~,. = E zn for all
n =O
z E D 1 (O). If we set k = 1 in Theorem 4.16, part (ii), then f' (z) = (1~>) 2 =
00 00E nzn- I = E (n + 1) zn, for all z E D1 (O).
n=I n=O- EXAMPLE 4. 25 The Bessel function of order zero is defined by