1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 535

7. To expand f(z) =Log z in powers of z - (-2 + i). This function is

analytic in D = C-{x: x:::; O}. For z E D we have

f'(z)=z-^1 , J"(z)=(-l)z-^2 , ... , f(n)(z)=(-l)n-^1 (n-l)!z-n

Hence·

f

(n)(-2 i) = (-l)n-l(n - 1)!

+ (-2 + i)n

(-l)n(n - 1)!(-2 - i)n (n - 1)!(2 + i)n

= =

and we get

.

00

Logz=Log(-2+i)-~ ~^1 (2+')n T (z+2-it

Although the series on the right converges on the disk D 1 = {z: lz -
( -2 + i) I < v'5}, it represents Log z only in the upper portion of D n D 1
(the shaded portion in Fig. 8.3). On the unshaded portion of D 1 the series
represents another branch of log z. The reader will note that the radius of
convergence of the series is the distance from the point a = -2 + i to the
branch point z = O, not the distance from -2+i to the set L = {x: x < O}
of the "artificial" singularities of Log z.
The proof of the Cauchy-Taylor theorem, as applied to the example
above, will require the substitution of the circle 01 by a simple closed
curve r about a and contained in D n D1; similarly for the case of some
other single-valued analytic branch of a multiple-valued function.

Remarks I. The real function f ( x) = 1 / ( 1 + x^2 ) is defined and has

derivatives of all orders for all real values of x. However, its Maclaurin

D

x

Fig. 8.3