1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

7.5 • APPLICATI ONS OF TAYLOR AND LAURENT SERIES 289

Laurent series for g (z) in the annulus D; (0) is

00

(-1)" 1

g(z) = 1 + 2::- 1 - 2n

n=l n. z

so that 0 is an essential singularity for g. According to Theorem 7.17, Jim lg (z)I
z- o
doesn't exist, so g is not continuous at 0.

-------~EXERCISES FOR SECTION 7.5

1. Determine whether there exists a function f that is analytic at 0 such that for

n = 1, 2, 3, ... ,

(a) f ( 2 ~) = 0 and f ( 2 ,.':_ 1 ) = 1.

(b) f (~) = f (";.1) = ;!or.

(c) /(~) =f(-;..^1 ) = ;!s-.

2. Prove the following corollaries and theorem.

(a) Corollary 7.9.

(b) Corollary 7.10.

(c) Theorem 7.14.

(d) Corollary 7.12.

3. Consider the function f (z) = zsin (~).
   (a) Show that there is a sequence {z,.} of points converging to 0 such that
   f (zn) = 0 for n = 1, 2, 3,. ...
   (b) Does this result contradict Corollary 7.9? Why or why not?


4. Let f (z) = tan z.
   (a) Use T heorem 7.14 to find the first few terms of the Maclaurin series
   for f (z) if lz l < ~·
   (b) What are the values off (^6 ) (0) and f (^7 ) (O)?


5. Show that the real function f defined by

f (x) = { xsin U) when x # 0, and

0 when x = 0,

is continuous at x = 0 but that the corresponding function g (z) defined by

g (z) = { zsin 0 (~) when when z z = # 0, 0, and

is not continuous at z = 0.