INDUS INSTITUTE OF ENGINEERING & TECHNOLOGY

Semester: IV Subject: COMPLEX ANALYSIS(MA0411)

UNIT-III
COMPLEX INTEGRATION

8 hours

1. Complex line integral: Properties and evaluation

2. Simply and multiply connected domains
3. Cauchys integral theorem, Cauchys integral theorem for multiply connected domains
4. Cauchys integral formula (without proof)

1 Complex line integral: Properties and evaluation

• Concept of integration as an inverse process of differentiation is also applicable in case of
complex functions provided,
Z the function is analytic.
i.e. If F 0 (z) = f (z), then f (z) dz = F (z) + c, where c is a complex constant.
This is called indefinite integral
Zz2
• The definite integral of a complex variable f (z) dz depends upon the path from z1 to z2 in
z1
the complex plane.
Z
• Complex definite integral can also be written as f (z) dz, where C is the path of integration.
C

• Complex definite integral is also called Complex Line Integral or simply line integral.
I
• When C is a closed path, i.e. when z1 and z2 co-incides, the integral denoted by f (z) dz.
C

• Properties of Line Integral:

Z Z
1. Let f (z) = u + iv and dz = dx + idy, then f (z) dz = (u + iv)(dx + idy)
Z Z C C

= (udx − vdy) + i (vdx + udy)

C C
2. Linearity Property:
Z Let k1 , k2 be complexZ constants andZ f (z) and g(z) be complex
functions, then [k1 f (z) + k2 g(z)] dz = k1 f (z) dz + k2 g(z) dz
C C C
Zz2 Zz1
3. f (z) dz = − f (z) dz
z1 z2

1
 Z Z
4. Given that C is an arc with end points A and B, |dz| = ds = L, where L is the
C C
length of the arc C from A to B.
Z Z Z
5. f (z) dz = f (z) dz + f (z) dz, where C is the curve consisting of two curves C1
C C1 C2
and C2 .
Zz2
• If f (z) is not analytic, value of f (z) dz is different for different paths from z1 to z2 . i.e. the
z1
value of the integral depends upon the path, when the function is not an analytic function.
Zz2
• When f (z) is an analytic function, f (z) dz is independent of the path from z1 to z2 .
z1

2 Simply and Multiply connected domains

• A path with coincident end points, not intersecting or touching itself is called a simple closed
path or contour. Integral the simple closed path is called contour integral.
• A domain D is called a simply connected domain, if every simple closed paths lying inside
D can be contracted to a point in D without leaving D.
A domain D which is not simply connected is called multiply connected domian.

• Fundamental Theorem of Complex Integration: If f (z) is analytic

I function in a simply
connected domain D, then for every simple closed path C in D, f (z) dz = 0.
C

• Evaluation of line integral: If f (z) be an analytic function in a simply connected domain

D, then there exists an analytic function F (z) with F 0 (z) = f (z) in D then along any path
Zz2
joining z1 and z2 in D, f (z) dz = F (z2 ) − F (z1 ) .
z1

• Cauchy’s Integral Theorem for Multiply Connected Domain: Let f (z) be analytic
between two simple closed paths C1 and C2 , where C2 lies entirely inside the curve C1 , then
I I
f (z) dz = f (z) dz
C1 C2

• Let C1 , C2 , . . . , Cn be finite simple closed paths inside a simple closed path C and f (z) is
analytic within the domain between the paths C1 , C2 , . . . , Cn then
I I I I
f (z) dz = f (z) dz + f (z) dz + · · · + f (z) dz
C C1 C2 Cn

2
3 Cauchy’s Integral Formula
• Let f (z) be an analytic function withinIand on a simple closed path C. If z0 is any point
f (z) f (z)
in C where is not analytic, then dz = 2πif (z0 )
z − z0 z − z0
C

• Generalized Cauchy’s integral formula: Let f (z) be an analytic function within and
f (z)
on a simple closed path C. If z0 is any point in C where is not analytic, then
z − z0
I
f (z) 2πi n
n+1
dz = f (z0 )
(z − z0 ) n!
C

• If f (z) is analytic on C1 and C2 and in the ring-shaped domain bounded by C1 and C2 and
f (z)
z0 is any point in that domain where is not analytic, then
z − z0
I I
f (z) f (z)
dz + dz = 2πif (z0 )
z − z0 z − z0
C1 C2

where the outer integral over C1 is taken counter clockwise and the inner integral over C2 is
taken clockwise.