Section 5: Complex Integration II
Examples and Problems
Example 1
[integration in the complex plane]
Evaluate the following integrals - mention whether the path of integration can be any arbitrary
contour between the points represented by the limits, or whether there are any restrictions on the
path:
j/2
e
(a)
z
dz
j
2j

(b)
2 j
1
dz
z3
Solution
(a)
The function f ( z )  ez is analytic everywhere so the path of integration can be any in the
complex plane. We have
j/2
e
j
z
dz 
1

j/2
e
z

j
e

1
j / 2

 e j 
1

cos( / 2)  j sin(  / 2)  cos( )  j sin(  )  (1  j ) / 
(b)
The function f ( z )  1 / z 3 is analytic everywhere except at the point z  0 , and the integral of
1 / z 3 ,  3 / z 2 , is also analytic everywhere except at z  0 . Thus we can choose as our path of
integration any path which does not pass through the origin. We have
2j
2j
 1
1
3
1  3
dz




3



3
2
2
2 j z
z 2 j
(2 j ) 2  2
 (2 j )
Example 2
[Cauchy's Integral Formula]
Integrate the function
z3
C 2 z  j dz
around the circle z  1.
Complex Variables Section 5 Examples & Problems - 1
Solution
The integrand has a singular point at z  j / 2 . This point lies within the closed contour C, so the
integrand z 3 /( 2 z  j ) is not analytic on or inside the closed contour C (so we can't use Cauchy's
Integral Theorem). We rewrite the integral in the form
1
z3
dz
2 C z  j / 2
which is the form of Cauchy's Integral Formula. We let f ( z )  z 3 , which is analytic on and
inside C, and z 0  j / 2 , so that f ( z 0 )  ( j / 2) 3   j / 8 . Thus
1
z3
1
dz  2j   j / 8   / 8

2C z  j/2
2
Example 3
[Cauchy's Integral Formula]
Integrate the function
z
C
dz
1
2
around the circle z  1  1.
Solution
The integrand has singular points at z  1 . We have
dz
C ( z  1)( z  1)
Only the singular point z  1 lies within the closed contour of integration, so we let z 0  1
and f ( z )  1 /( z  1) :
dz
f ( z)
C z 2  1  C z  1 dz
Now f ( z 0 )  1 /(1  1)  1 / 2 . Thus

C
f ( z )dz
dz  2j  1 / 2  j
z 1
Example 4
[Formulas for derivatives of an analytic function]
Integrate the function
z3
dz
3

C ( z  1)
where C is any closed contour enclosing the point z  1 .
Complex Variables Section 5 Examples & Problems - 2
Solution
We can write this in the form
f ( z)
dz
3
0)
 (z  z
C
where
f ( z )  z 3 , z 0  1
The function f (z ) is analytic on and inside the contour C so we can apply the formula
f ( z)
 z  z 
C
0
n 1
dz 
2j d n f
n! dz n
Setting n  2 puts the formula in the form we require:
f ( z)
2j d 2 f
dz

3

2! dz 2
C z  z 0 
z0
z0
Now
d2 3
d2 f
(
z
)

6
z

dz 2
dz 2
so that
f ( z)
 z  z 
3
C
 6
z 0  1
dz  6j
0
Complex Variables Section 5 Examples & Problems - 3
Problem 1
[Integration in the Complex Plane]
(i)
Use Cauchy's Integral Theorem to show that, if f (z ) is analytic in a simply connected
domain D, and z 0 and z1 are any two points in D, the integral
z1
 f ( z )dz
z0
(ii)
is independent of the path taken (so long as it remains in D).
Evaluate the following integrals
 2 j
 cos( z / 2)dz
(a)
[ANSWER: e  1/ e ]
0
3
(b)
 ( z  2)
3
dz
[ANSWER: 0 ]
1
3j
(c)
 e
2z
[ANSWER: 0 ]
dz
 j
Problem 2
[Cauchy's Integral Formula]
(i)
Under what conditions is Cauchy's Integral Formula valid?
cos z
(ii)
Evaluate the integral 
dz where C is z  1 [ANSWER: j ]
2
z
C
tan z
(iii) Evaluate the integral 
dz counterclockwise around the triangle with vertices
z j
C
z  1 , z  2 j and z  1 . Draw the triangle in the complex plane and show clearly the
positions of the singular point(s) of the integrand. [ANSWER:  2 tanh( 1)  4.785 ]
2z  1
(iv)
Evaluate the integral  2
dz where C is
C z  z
(a) z 
(v)
1
,
4
(b) z 
1 1
 ,
2 4
(c) z  2 [ANSWER: (a) 2j , (b) 0, (c) 4j ]
ez
dz where C is z  2 [ANSWER: 5.287 j]
Evaluate the integral  2
C z 1
Problem 3
[Formulas for derivatives of an analytic function]
z4
cos z
dz ,  2 dz , and
Show that the three integrals 
2
C ( z  3 j)
C z
are all zero.
3
ez
 3 dz , where C is the unit circle,
C z
Complex Variables Section 5 Examples & Problems - 4