Complex Analysis - Maths KU

386 Chapter 6 Series and Residues
Projects
26. In the application of the Laplace transform to problems involving partial differential
equations, one often encounters an inverse such as
f(x, t) =� −1
�
sinh xs
(s2 �
.
+ 1) sinh s
Investigate how (8) and (9) can be used to determine f (x, t).
Answers to selected odd-numbered problems begin
CHAPTER 6 REVIEW QUIZ
on page ANS-21.
In Problems 1–20, answer true or false. If the statement is false, justify your answer
by either explaining why it is false or giving a counterexample; if the statement is
true, justify your answer by either proving the statement or citing an appropriate
result in this chapter.
1. For the sequence {zn}, where zn = i n = xn + iyn, Re(zn) =xn = cos(nπ/2)
and Im(zn) =yn = sin(nπ/2).
2. The sequence { i n } converges.
� �n 1+i
3. limn→∞ √ =0.
π
4. limn→∞ zn = 0 if and only if limn→∞ |zn| =0.
5. The power series �∞ convergence.
k=1
z k
converges absolutely at every point on its circle of
k2 6. There exists a power series centered at z0 =1+i that converges at z =25−4i and diverges at z =15+21i.
7. A function f is analytic at a point z0 if f can be expanded in a convergent
power series centered at z0.
8. Suppose a function f has a Taylor series representation with circle of convergence
|z − z0| = R, R>0. Then f is analytic everywhere on the circle of
convergence.
9. Suppose a function f has a Taylor series representation centered at z0.
Then f is analytic everywhere inside the circle of convergence |z − z0| = R,
R>0, and is not analytic everywhere outside |z − z0| = R.
10. If the function f is entire, then the radius of convergence of a Taylor series
expansion of f centered at z0 =1− i is necessarily R = ∞.
11. Both power series
1
1+z =1− z + z2 − z 3 + ···
and
converge at z =0.86 − 0.52i.
1 1 z − 1
= −
1+z 2 22 (z − 1)2
+
23 (z − 1)3
−
24 + ···