Complex Analysis - Maths KU

Chapter 6 Review Quiz 387
12. If the power series �∞ k=0 akz k has radius of convergence R, then the power
series �∞ k=0 akz 2k has radius of convergence √ R.
13. The power series �∞ k=0 akz k and �∞ k=1 kakz k−1 have the same radius of convergence
R.
14. The principal branch f1(z) of the complex logarithm does not possess a Maclaurin
expansion.
15. If f is analytic throughout some deleted neighborhood of z0 and z0 is a pole of
order n, then limz→z0(z − z0) n f(z) �= 0.
16. A singularity of a rational function is either removable or is a pole.
1
17. The function f(z) =
z2 , a>1, has two simple poles within the unit
+2iaz − 1
circle |z| =1.
18. z = 0 is a simple pole of f(z) =− 1
+ cot z.
z
19. If z0 is a simple pole of a function f, then it is possible that Res(f(z),z0) =0.
1
20. The principal part of the Laurent series of f(z) =
valid for
1 − cos z
0 < |z| < 2π contains precisely two nonzero terms.
In Problems 21–40, try to fill in the blanks without referring back to the text.
� �
2in (9 − 12i)n +2
21. The sequence − converges to
n + i 3n +1+7i
.
22. The series i +2i +3i +4i + ··· diverges because .
23. 5 − i − 1 i 1
+ + −··· =
5 25 125
.
24. The equality � � �k ∞ z − 1
=
k=0 z +1
1
(z + 1) comes from 2 and is valid
in the region of the complex plane defined by .
25. The power series �∞ k=0 (5 + 12i)k (z − 2 − i) k converges absolutely within the
circle .
26. The power series �∞ 4
k=0
k
2k +5 (z − 2+3i)2kdiverges for | z − 2+3i | >
27. If the power series
.
�∞ k=0 akz k , ak �= 0, has radius of convergence R>0, then
the power series �∞ z k
has radius of convergence .
k=0 ak
28. Without finding the actual expansion, the Taylor series of f(z) = csc z centered
at z0 =3+2ihas radius of convergence R = .
29. Use the first series in Problem 11 to obtain the first three terms of a Taylor series
z +1
of f(z) = centered at z0 = −1: . The radius of convergence of
6+z
the series is R = .
30. A power series centered at −5i for f(z) =e z is given by e z = �∞ (z +5i) k .
k=0