Complex Analysis - Maths KU

322 Chapter 6 Series and Residues
29. What is the radius of convergence R of the Maclaurin series in Problem 23?
30. What is the radius of convergence R of the Maclaurin series in Problem 24?
In Problems 31 and 32, expand the given function in Taylor series centered at each
of the indicated points. Give the radius of convergence R of each series. Sketch the
region within which both series converge.
31. f(z) = 1
2+z
1
, z0 = −1, z0 = i 32. f(z) = , z0 =1+i, z0 =3
z
In Problems 33 and 34, use results obtained in this section to find the sum of the
given power series.
33. ∞�
3 k z k
34. ∞� z 2
k!
k=0
35. Find the Maclaurin series (14) by differentiating the Maclaurin series (13).
36. The error function erf(z) is defined by the integral erf(z) = 2
√ π
k=0
� z
0
e −t2
dt .
Find a Maclaurin series for erf(z) by integrating the Maclaurin series for e −t2
.
In Problems 37 and 38, approximate the value of the given expression using the
indicated number of terms of a Maclaurin series.
37. e (1+i)/10 � �
1+i
, three terms 38. sin , two terms
10
Focus on Concepts
39. Every function f has a domain of definition. Describe in words the domain of
the function f defined by a power series center at z0.
40. If f(z) = � ∞
k=0 akz k and g(z) = � ∞
k=0 bkz k then the Cauchy product of f
and g is given by
f(z)g(z) =
∞�
k�
ckz
k=0
k where ck = anbk−n.
n=0
Write out the first five terms of the power series of f(z)g(z).
41. Use Problem 40, (12) of this section, and (6) from Section 6.1 to find the first
four nonzero terms of the Maclaurin series of e z /(1 − z). What is the radius of
convergence R of the series?
42. Use Problem 40, and (13) and (14) of this section to find the first four nonzero
terms of the Maclaurin series of sin z cos z. Can youthink of another way to
obtain this series?
43. The function f(z) = sec z is analytic at z = 0 and hence possesses a Maclaurin
series representation. We could, of course, use (7), but there are several
alternative ways of obtain the coefficients of the series
sec z = a0 + a1z + a2z 2 + a3z 3 + ··· .