Complex Analysis - Maths KU

6.5 Residues and Residue Theorem 351
29.
30.
31.
32.
33.
34.
�
�
�
�
�
�
C
cot πz dz, C is the rectangle defined by x = 1
, x = π, y = −1, y =1
2
2z − 1
C z2 (z3 +1)
C
C
C
C
�
z 2 e 1/πz + zez
z4 − π4 �
1
dz, C is the rectangle defined by x = −2, x =1,y = − , y =1
2
dz, C: 4x 2 + y 2 =16
cos z
(z − 1) 2 (z2 dz, C: |z − 1| =1
+9)
1
z 6 +1 dz, C is the semicircle defined by y =0,y = √ 4 − x 2
e 4/(z−2) dz, C: |z − 1| =3
Focus on Concepts
35. (a) Use series to show that z = 0 is a zero of order 2 of 1 − cos z.
(b) In view of part (a), z = 0 is a pole of order two of the function
f(z) =e z /(1 − cos z) and hence has a Laurent series
f(z) =
e z
1 − cos z
a−2 a−1
= +
z2 z + a0 + a1z + a2z2 + ···
valid for 0 < |z| < 2π. Use series for e z and 1−cos z and equate coefficients
in the product
e z � �
a−2 a−1
=(1−cos z) + + a0 + ···
z2 z
to determine a−2, a−1, and a0.
�
e
(c) Evaluate
C
z
dz, where C is |z| =1.
1 − cos z
�
36. Discuss how to evaluate e
C
1/z � �
1
sin dz, where C is |z| = 1. Carry out your
z
ideas.
37. Consider the function f(z) =z 4
�
(1 − z 1/2 ), where z 1/2 denotes the principal
branch of the square root function. Discuss and justify your answer: Does f
have a pole at z = 1? If so, find Res(f(z), 1).
38. Residues can be used to find coefficients in partial fraction decompositions of
rational functions. Suppose that p(z) is a polynomial of degree ≤ 2 and that
z1, z2, and z3 are distinct complex numbers that are not zeros of p(z). Then
f(z) =
(a) Use (1) to show
p(z)
A
= +
(z − z1)(z − z2)(z − z3) z − z1
B
+
z − z2
C
.
z − z3
p(z1)
A = Res(f(z),z1) =
(z1 − z2)(z1 − z3)
p(z2)
B = Res(f(z),z2) =
(z2 − z1)(z2 − z3)
p(z3)
C = Res(f(z),z3) =
(z3 − z1)(z3 − z2) .