Complex Analysis - Maths KU

6.6 Some Consequences of the Residue Theorem 373
57. Suppose a real function f is continuous on the interval [a, b] except at a point
c within the interval. Then the principal value of the integral is defined by
� b
�� c−ε � b �
P.V. f(x) dx = lim
ε→0
f(x) dx + f(x) dx , ε > 0.
a
� 3
1
Compute the principal value of
x − 1 dx.
58. Determine whether the integral in Problem 57 converges.
0
a
6.6.4 The Argument Principle and Rouché’s Theorem
In Problems 59 and
�
60, use the argument principle in (28) of Theorem 6.20 to
f
evaluate the integral
C
′ (z)
dz for the given function f and closed contour C.
f(z)
59. f(z) =z 6 − 2iz 4 +(5−i)z 2 + 10, C encloses all the zeros of f
60. f(z) =
(z − 3iz − 2)2
z(z2 3
, C is |z| =
− 2z +2) 5 2
In Problems 61–64, use the argument principle in (28) of Theorem 6.20 to evaluate
the given integral on the indicated closed contour C. Youwill have to identify f(z)
and f ′ (z).
�
2z +1
61.
C z2 dz, C is |z| =2
+ z
62.
�
z
C z2 �
dz, C is |z| =3
+4
63. cot zdz, C is the rectangular contour with vertices 10 + i, −4 +i, −4 − i,
C
and 10 − i.
�
64. tan πz dz, C is |z − 1| =2
C
65. Use Rouché’s theorem (Theorem 6.21) to show that all seven of the zeros of
g(z) =z 7 +10z 3 + 14 lie within the annular region 1 < |z| < 2.
66. (a) Use Rouché’s theorem (Theorem 6.21) to show that all four of the zeros of
g(z) =4z 4 + 2(1 − i)z + 1 lie within the disk |z| < 1.
(b) Show that three of the zeros of the function g in part (a) lie within the
< |z| < 1.
annular region 1
2
67. In the proof of Theorem 6.21, explain how the hypothesis that the strict inequality
|f(z) − g(z)| < |f(z)| holds for all z on C implies that f and g cannot
have zeros on C.
6.6.5 Summing Infinite Series
68. (a) Use the procedure illustrated in Example 8 to obtain the general result
∞� 1
k2 1 π
= + coth aπ.
+ a2 2a2 2a
k=0
(b) Use part (a) to verify (47) when a =2.
∞� 1
(c) Find the sum of the series
k2 +1 .
k=0
c+ε