Complex Analysis PH 503 CourseTM
Charudatt Kadolkar Indian Institute of Technology, Guwahati
ii
Copyright
2000 by Charudatt Kadolkar
Preface Preface Head These notes were prepared during the lectures given to MSc students at IIT Guwahati, July 2000 and 2001..
Acknowledgments
As of now none but myself IIT Guwahati
Charudatt Kadolkar.
Contents
Preface
iii Preface Head
iii
Acknowledgments
1
Complex Numbers
iii
1
De• nitions
1
Algebraic Properties
1
Polar Coordinates and Euler Formula Roots of Complex Numbers Regions in Complex Plane
2
Functions of Complex Variables
3 3
5
Functions of a Complex Variable Elementary Functions Mappings
5
5
7
Mappings by Elementary Functions.
3
Analytic Functions Limits
2
11 11
Continuity
12
Derivative
12
Cauchy-Riemann Equations
13
8
vi
Contents Analytic Functions
14
Harmonic Functions
4
Integrals
14
15 Contours
15
Contour Integral
16
Cauchy-Goursat Theorem Antiderivative
17
Cauchy Integral Formula
5
Series
17
18
19 Convergence of Sequences and Series Taylor Series Laurent Series
6
20 20
Theory of Residues And Its Applications Singularities
23
Types of singularities Residues
23
24
Residues of Poles
24
Quotients of Analytic Functions
A
References
B
Index
29
27
23
25
19
1
Complex Numbers
De• nitions De• nition 1.1 Complex numbers are de• ned as ordered pairs
Points on a complex plane. Real axis, imaginary axis, purely imaginary numbers. Real and imaginary parts of complex number. Equality of two complex numbers. De• nition 1.2 The sum and product of two complex numbers are de• ned as follows:
!
"
#
$
%
In the rest of the chapter use introduce and notation. .
6
7
8
9
:
6
/
.
0
/
.
1
/
2
&
'
(
'
)
*
+
(
'
)
,
+
for complex numbers and
2
2
)
3
'
(
4
-
for real numbers. 5
;
Algebraic Properties
1. Commutativity 7
<
:
7
=
8
7
=
:
7
<
>
7
<
7
=
8
7
=
7
<
2. Associativity ?
7
<
:
7
=
@
:
7
A
B
C
D
E
F
C
G
E
C
A
H
I
F
C
D
C
G
H
C
A
B
3. Distributive Law
C
C
F
C
D
E
C
G
H
B
C
C
D
E
C
C
G
4. Additive and Multiplicative Indentity C
E
J
K
L
M
L
N
O
P
Q
5. Additive and Multiplicative Inverse R
R
Q
P
T
U
R
V
W
S
[
\
`
g
Q
X
Y
h
Z
]
^
_
`
^
a
]
^
b
_
`
^
c
d
e
f
D
F
C
G
C
A
H
2
Chapter 1 Complex Numbers 6. Subtraction and Division i
j
j
i
j
k
i
l
m
i
j
n
o
k
i
l
p
q
m
i
j
i
r
l
i
7. Modulus or Absolute Value s
8. Conjugates and properties {
t
s
{
u
v
|
w
x
w
y
}
z
x
|
~
z
l
w
}
z
9.
10. Triangle Inequality
¡
¢
Polar Coordinates and Euler Formula
£
1. Polar Form: for ¤
, ¥
¤
¦
§
¨
©
ª
«
¬
®
¯
°
±
²
where and . is called the argument of . Since an argument of the principle value of argument of is take such that For the is unde• ned. ³
´
µ
¶
µ
·
Å
Ë
¸
¹
º
»
¼
½
¾
¿
À
Æ
Å
Ì
Í
Î
Ï
Ð
Ñ
Ò
Ó
2. Euler formula: Symbollically, Ô
3. â
Õ
ã
Ö
â
×
Ø
Ù
Ú
Û
Ü
ä
Ý
Þ
ã
×
ä
å
Û
æ
ç
ß
è
à
é
Ü
ê
ë
á
ì
í
î
å
î
ï
ð
ï
ð
ñ
ò
ó
ñ
ô
õ
ö
÷
ø
ù
÷
ú
ó
ï
4. de Moivre’s Formula
û
ü
ý
û
þ
û
û
ü
ÿ
¿
Á
Â
Ç
Ã
is also Ä
Ä
È
É
Ê
Roots of Complex Numbers
3
Roots of Complex Numbers Let
then
"
!
#
#
There are only distinct roots which can be given by principle value of then is called the principle root. "
"
$
%
&
'
&
(
(
(
If )
&
'
is a *
(
"
+
,
-
.
*
/
Example 1.1 The three possible roots of
4
0
1
5
2
6
7
8
9
:
;
<
=
>
?
@
A
B
3
C
D
are E
F
G
E
C
B
Regions in Complex Plane
1. L
M
N
O
of P
Q
R
is de• ned as a set of all points which satisfy S
T
S
T
2. X
Y
Z
[
\
]
^
_
`
a
of b
c
is a nbd of b
c
U
S
V
W
R
excluding point b
c
.
3. Interior Point, Exterior Point, Boundary Point, Open set and closed set. 4. Domain, Region, Bounded sets, Limit Points.
H
I
F
G
F
C
B
H
J
G
H
E
C
D
I
F
G
F
C
B
H
J
K
G
C
D
.
2
Functions of Complex Variables
Functions of a Complex Variable A de• ned on a set is a rule that uniquely associates to each point of a a complex number Set is called the of and is called the value of at and is denoted by d
e
f
g
h
i
j
k
l
m
o
p
x
x
{
z
|
}
x
{
|
{
n
q
z
|
}
r
}
y
s
t
u
v
w
x
m
y
x
z
~
Example 2.1 Write
ª
«
¬
®
¯
°
¬
in ±
²
³
´
¡
¢
£
¤
¥
¦
§
¨
©
form. µ
¿
¾
²
«
¶
·
¸
and
¿
®
¹
º
»
¼
½
¾
Â
Ã
Ä
Å
Æ
Ç
¿
È
É
¼
¼
¼ ½
½
À
½
½
Á
½
and
Ê
Ã
Ë
Å
Ì
Ç
È
Ë
Í
Ì
É
Domain of A domain. Û
Ü
Ý
Þ
ß
à
á
â
Î
Ï
is
Ð
Õ
ã
ä
å
æ
Ñ
Ö
ç
×
è
é
Ø
ê
Ù
ç
Ã
Ë
Å
Ì
Ç
È
Ò
Ë
½
Á
Í
Ì
É
Ð
Ó
Ô
Ñ
. Ú
ë
Â
À
ì
í
î
ï
is a rule that assigns more than one value to each point of ë
Example 2.2 This function assigns two distinct values to each One can choose the function to be single-valued by specifying ð
ñ
ò
ó
ô
õ
ö
÷
ú
õ
where is the principal value.
Elementary Functions
ý
õ
ö
þ
ÿ
ú
ö
ø
û
ù
ü
.
6
Chapter 2 Functions of Complex Variables 1. Polynomials
whrere the coef• cients are real. Rational Functions. 2. Exponential Function
Converges for all For real tion. Easy to see that "
#
$
%
&
'
!
!
!
the de• nition coincides with usual exponential func. Then
"
(
)
*
+
*
-
.
+
,
$
/
'
$
0
1
(
)
*
2
*
-
.
2
3
,
a. . b. c. A line segment from origin. d. No Zeros. $
$
/
4
/
6
'
/
$
5
/
$
%
7
'
4
6
$
/
5
#
8
/
1
9
:
;
to 3
1
9
:
<
maps to a circle of radius
=
3
$
0
centered at
3. Trigonometric Functions De• ne $
%
$
>
%
/
(
)
*
"
/
'
<
$
%
$
/
*
-
.
"
>
%
?
/
'
<
*
@
-
.
"
a. b. c. d. e. and f. iff g. iff h. These functions are not bounded. i. A line segment from to equal to under function. 7
-
.
"
(
)
*
"
'
"
'
(
*
.
A
)
*
"
B
7
<
*
-
.
"
C
(
)
*
"
'
*
-
.
1
"
C
"
3
7
<
(
<
*
)
*
"
(
C
)
*
"
'
(
)
*
1
"
"
.
"
C
*
-
.
-
3
C
7
-
*
.
1
"
C
"
(
7
)
*
1
"
" C
'
(
)
*
1
"
C
"
7
3
7
.
*
-
.
(
-
1
)
"
*
"
<
"
'
=
3
(
)
*
1
"
C
"
'
*
"
;
3
?
7
'
;
3
?
7
"
?
*
3
?
7
'
"
-
.
"
D
(
=
)
1
'
D
*
D
1
"
<
'
=
;
1
:
7
=
E
3
D
B
'
'
:
(
;
#
:
#
#
E
)
*
"
#
3
B
:
#
#
#
3
8
7
1
(
)
*
F
G
H
;
I
:
2
3
1
<
=
:
2
maps to an ellipse with semimajor axis 3
J
4. Hyperbolic Functions De• ne P
K
Q
P
L
H
F
M
N
O
O
R
S
H
P
I
J
F
M
N
T
O
P
O S
R
Mappings a. b. c. d. e. U
V
W
_
X
`
Y
a
b
a
j
k
b
a
j
k
b
_
`
Z
h
[
\
c
e
b
Z
i
a
m
e
a
]
e
n
j
g
o
g
V
p
W
b
d
[
h
f
^
e
a
e
_
g
g
iff iff
p
e
U
k
a
b
c
d
e
f
g
_
`
a
e
l
j
k
g
e
`
7
b
q
g
c
o
u
e
f
^
_
d
c
v
m
q
o
q
`
a
g
w
b
c
p
r
e
d
s
m
l
c
n
r
q
t
o
t
g
d
p
t
f
g
_
`
a
b
c
e
f
f
s r
l
r
t
t
t
f
h
5. Logarithmic Function De• ne then
x
y
z
{
|
x
y
z
}
~
~
a. Is multiple-valued. Hence cannot be considered as inverse of exponetial function. b. Priniciple value of log function is given by
where c.
is the principal value of argument of .
Mappings . Graphical representation of images of sets under cally shown in following manner:
is called
¡
¢
£
£
¤
¥
¦
Typi§
1. Draw regular sets (lines, circles, geometric regions etc) in a complex plane, which we call plane. Use ¨
¨
©
ª
«
¬
©
®
¯
°
±
²
2. Show its images on another complex plane, which we call . ·
«
¬
¸
©
Example 2.3
¹
º
½
»
¼
¾
¿
À
.
³
plane. Use ³
©
´
µ
¨
¶
©
8
Chapter 2 Functions of Complex Variables 2
1.5
1
0.5
-0.6-0.4-0.2 0 0 0.2 0.4 0.6 0.8 1
Mapping of 1. A straight line 2. A straight line
Ã
Ä
Ñ
Mapping Â
maps to a parabola Å
Ò
Á
Ë
maps to a parabola
Æ
Ç
Ó
È
Ì
É
Ò
Ê
Ê
Ë
Ë
Ì
Ì
Í
Í
Î
Î
Ï
Ï
Ë
Ë
Ì
Ì
Á
Â
Ð
Ð
3. A half circle given by where maps to a full circle given by This also means that the upper half plane maps on to the entire complex plane. Ô
Ò
Õ
Ö
×
Ø
Ù
Ú
Û
Ü
Û
Ý
Þ
Ò
Õ
Ì
×
Ø
Ì
Ù
ß
Ö
4. A hyperbola à
Ì
Ñ Ï
Ì
Ò
á
maps to a straight line â
ã
Mappings by Elementary Functions.
á
1. Translation by å
æ
is given by
2. Rotation through an angle é
æ
ç
ã
å
è
æ
is given by
3. Rel• ection through x axis is given by 4. Exponential Function
å
ä
ç
ã
î
ç
ã
í
ï
ê
ë
ì
å
ä
ä
Mappings by Elementary Functions.
9
Exponential Function A vertical line maps to a circle. A horizontal line maps to a radial line. A horizontal strip enclosed between plane. 5. Sine Function õ
ö
÷
ø
ñ
õ
ö
÷
ù
ú
û
õ
ü
ð
ý
ð
þ
ñ
and
ú
ò
û
õ
ù
õ
ö
÷
ð
ü
ñ
ó
maps to the entire complex ô
ð
A vertical line maps to a branch of a hyperbola. A horizontal line maps to an ellipse and has a period of ó
ô
.
3
Analytic Functions
Limits A function
is de• ned in a deleted nbd of ÿ
De• nition 3.1 The limit of the function given there exists a such that
+
,
-
.
Example 3.2
Example 3.3 +
,
"
-
3
-
/
!
.
Theorem 3.1 Let
/
+
-
,
-
9
.
$
Show that 2
-
:
is a number
%
4
&
5
'
6
(
7
)
8
*
7
+
*
,
+
-
,
.
-
/
=
,
>
?
@
.
/
.
4
E
F
G
H
5
6
=
I
E
F
G
8
Example 3.4 Example 3.5
H
/
=
A
B
and 1
I
*
C
,
>
4
E
F
G
H
?
,
-
.
/
+
,
-
.
/
5
@
0
/
-
-
1
3
2
2
does not exist as +
and .
6
D
C
I
E
*
+
F
G
8
J
5
K
L
A
B
@
H
1
/
=
1
A
B
C
1
2
4
C
1
4
7
5
8
6
+
7
,
-
.
/
D
and 1
4
7
S
V
T
5
4
5
6
8
*
6
8 7
,
7
*
6
+
7
. <
,
-
.
/
D
1
if
*
2
7
4
5
+ *
6
,
-
-
.
.
8 7
/
/
+
3
Theorem 3.2 If
5
8
;
*
. Show that the 3
/
-
I
*
. Show that the -
>
if, for any
1
Show that the limit of 2
7
and only if
Show that #
Example 3.1
as
ÿ
J
,
-
.
5
K
-
/
1
N
B
.
P
M
Q
R
O
U
W
V
X
Y
g
j
g
Z
h
[
k
h
\
]
^
_
[
\
]
`
a
b
c
b
c
k
e
c
f
j
l
m
n
o
p
q
n
o
p
a
q
n
o
p
a
e
c
f
i
b
j
d
i
j
l
m
n
o
p
r
c
e
e
c
a
s
t
u
c
This theorem immediately makes available the entire machinery and tools used for real analysis to be applied to complex analysis. The rules for • nding limits then can be listed as follows:
12
Chapter 3 Analytic Functions 1. v
w
y
x
z
2.
y
{
|
{
{
}
|
~
v
w
x
z y
3.
y
}
~
if
v
w
y
x
z
4.
y
}
is a polynomial in .
v
w
x
z y
5.
{
y
}
~
v
y
w
x
w
z y
{
w
}
~
Continuity De• nition 3.2 A function , de• ned in some nbd of
is continuous at
if
v
w
x
z y
y
{
}
~
This de• nition clearly assumes that the function is de• ned at and the limit on the LHS exists. The function is continuous in a region if it is continuous at all points in that region.
If funtions and continuous at .
are continuous at
then
,
and
are also
If a function and are also continuous at
is continuous at
then the component functions
.
Derivative De• nition 3.3 A function , de• ned in some nbd of
¢
Example 3.7 real analysis
¬
¯
±
²
Show that ³
°
is differentiable at
if
¡
exists. The limit is called the derivative of Example 3.6
¬
°
¢
at
¯
±
and is denoted by
µ
£
¤
¥
¦
§
or ª
«
¨
¬
®
©
³
. Show that this function is differentiable only at is not differentiable but is.
¹
º
¹
¯
±
¶
¶
¹
º
¹
»
If a function is differentiable at , then it is continuous at . ¼
.
´
¸
²
¬
°
¯
¼
±
·
. In
Cauchy-Riemann Equations
13
The converse in not true. See Example 3.7. Even if component functions of a complex function have all the partial derivatives, does not imply that the complex function will be differentiable. See Example 3.7. Some rules for obtaining the derivatives of functions are listed here. Let differentiable at ¿
Ã
1.
½
Ä
¾
Ã
Å
and
be ¾
À
Ã
Á
Á
½
¿
Å
Æ
½
Ã
Ç
¿
Å
Ä
¾
¿
Å
À
Â
Ã
2.
Ã
Á ½
Á
¾
Ã
Å
¿
Å
Æ
½
Ã
Ç
¿
Å
Ã
¾
¿
Å
È
Ã
½
¿
Å
¾
Ç
¿
Å
À
Â
Ã
3.
Ã
Á
½
Á
É
¾
Ã
Å
¿
Å
Ã
Æ
½
Ã
¿ Ç
Å
Ã
¾
¿
Å
Ê
Ã
½
¿
Å
¾
Ã
¿ Ç
Å
Å
É
Ë
Ã
¾
¿
Å
Ì
if Í
¾
¿
Å
Æ
Î
Ï
À
Â
Ã
4. Á
½
Á
Ð
Ñ
Ò
Ó
Ô
Ò
Õ
Ö
×
Ó
Ñ
Ó
Ô
Ò
Ò
Ñ
×
Ó
Ô
Ò
Ø
Â
5. Ù
Õ
Ù
6.
Ú
Ù
Ù
Ü
Ø
Û
Ô
Ý
Õ
Þ
Ô
Ý
ß
à
Ø
Ú
Cauchy-Riemann Equations Theorem 3.3 If function and ×
Ö
â
Ó
ã
ä
å
Ó
Ô
Ò
á
exists, then all the • rst order partial derivatives of component exist and satisfy Cauchy-Riemann Conditions: Ò
æ
Ó
ã
ä
å
Ò
â
ç
â
Õ
Example 3.8 are satis• ed. Example 3.9 satis• ed only at
Ö
Ó
Ô
Ò
Õ
Ô
ê
Õ
é
ã
ê
Õ è
å
ê
Ó
Ô
Ô
Ò
Õ
Õ
î
Ô
î
Õ
ã
ë ê
å
ì
í
ã
æ
å
Ø ç
Show that Cauchy-Riemann Condtions Ø
Show that the Cauchy-Riemann Condtions are Ø
ê
è
é
ë
ê
Ö
æ
. Ü
Theorem 3.4 Let be de• ned in some nbd of the point If the • rst partial derivatives of and exist and are continuous at and satisfy CauchyRiemann equations at , then is differentiable at and Ö
Ó
Ô
Ò
Õ
â
Ó
ã
ä
å
Ò
ë
â
Ô
ì
æ
Ó
ã
ä
å
Ò
Ô
æ
á
Ô
Ö
Ô
á
Ø
á
á
×
Ö
Example 3.10
Ó
Ô
Ò
Õ
Ö
Ó
Ô
Ò
Õ
Example 3.11
Ö
ï
ð
ò
ñ
ó
ø
Ó
ô
Ó
Ó
Ô
Ô
Ô
Ò
Ò
Ò
Õ
Ø
â ç
ë
ì
Õ ç
Show that Show that
Ø
æ
Ö
Ö
×
×
Ó
Ó
é æ
Ô
Ô
Ò
Ò
ì
è
Õ
Õ
õ
ï
â
ö
ð
Ø è
ò
ñ
Ó
Ó
Ô
Ò
Ô
Ò
Ø
Ø
ù
Example 3.12 Show that the CR conditions are satis• ed at function is not differentiable at Ö
Ó
Ô
Ò
Õ
÷
þ
ø
ú
ý
ú
û
ü
ý
but the
14
Chapter 3 Analytic Functions If we write nates:
then we can write Cauchy-Riemann Conditions in polar coordi
ÿ
Analytic Functions De• nition 3.4 A function is analytic in an open set if it has a derivative at each point in that set. De• nition 3.5 A function is analytic at a point ÿ
if it is analytic in some nbd of
De• nition 3.6 A function is an entire function if it is analytic at all points of Example 3.13
Example 3.14
ÿ
ÿ
ÿ
ÿ
.
is analytic at all nonzero points.
ÿ
is not analytic anywhere.
A function is not analytic at a point but is analytic at some point in each nbd of then is called the singular point of the function . ÿ
ÿ
ÿ
Harmonic Functions De• nition 3.7 A real valued function is said to be in a domain of xy plane if it has continuous partial derivatives of the • rst and second order and satis• es :
"
#
$
%
&
'
(
(
)
*
+
,
-
.
!
/
0
1
2
1
=
2
6
3
4
9
5
6
>
7
2
Theorem 3.5 If a function the functions and are harmonic in
0
3
4
8
5
6
2
3
7
4
?
5
6
@
2
9
3
4
:
5
;
6
is analytic in a domain
<
B
C
then A
. A
De• nition 3.8 If two given functions and are harmonic in domain D and their • rst order partial derivatives satisfy Cauchy-Riemann Conditions B
B
C
N
O
Example 3.15 Let not vice versa. Example 3.16 \
^
_
P
\
Q
`
^
a
R
_
S
b
`
c
a
T
U
b
U
a
c
R
h
S
e
_
V
i
d
W
X
_
e
Y
F
G
H
J
K
d
[
C
C
J
Z
a
a
E
I
B
then is said to be
D
of
and
E
F
G
H
K
C
I
L
\
D
M
f
]
^
_
`
a
b
c
g
_
a
]
Show that is hc of
. Find harmonic conjugate of \
f
]
\
and
4
Integrals
Contours Example 4.1 Represent a line segment joining points equations. j
k
l
m
and n
o
p
q
r
r
Example 4.2 Show that a half circle in upper half plane with radius at origin can be parametrized in various ways as given below: 1.
u
2.
t
3.
v
w
x
s
y
z
{
v
|
}
v
u
w
x
s
and
~
where
¢
, where
¡
if
. Show that the
.
An arc is differentiable if ous. A smooth arc is differentiable and
and centered s
in complex plane is called an
v
are continuous functions of the real parameter
An arc is called simple if An arc is closed if
where |
where
v
Example 4.3 curve cuts itself and is closed.
De• nition 4.1 A set of points where
{
by parametric n
¢
£
¤
¢
¢
exists and and is nonzero for all .
¦
§
¨
©
ª
«
¯
¬
De• nition 4.2 Length of a smooth arc is de• ned as ¥
¢
°
±
²
³
´
µ
®
The length is invariant under parametrization change.
²
¶
¢
are continu-
16
Chapter 4 Integrals De• nition 4.3 A is a constructed by joining • nite smooth curves end to end such that is continuous and is piecewise continuous. ·
½
¾
¿
¸
¹
º
¸
»
¼
À
½
Á
¾
¿
À
A closed simple contour has only • rst and last point same and does not cross itself.
Contour Integral If
is a contour in complex plane de• ned by is de• ned on it. The integral of denoted and de• ned as follows: Â
½
¾
¿
À
Ã
Ä
¾
È
¾
½
À
Ã
É
¾
Ä
Ê
Ç
À
Å
Ì
Æ
Í
Ë
¾
Î
Ï
Ð
Ä
Ê
Ñ
Ç
Ò
¿
À
À
Ð
Ó
Ô
Ó
Ô
Ó
Ô
Õ
Î
Ï
Ð
Ñ
Ð
×
Ï
Ø
Ñ
Ò
and a function along the contour is
Å
Æ
È
¾
½
À
Ç
¾
¿
À
Â
Ø
Ö
Ï Õ
Ù
Ú ×
Û
Ü
Ý
×
Ñ
Ò
Ø
Þ
ß
Ô
Ï Õ
Ö
Ù Ý
Þ ×
Ú Ü
Ñ ×
Ò
Ø
Ö
Ï
Ù
Ò
Ú
Û
Ò Ü
Ñ Ý
Þ
ß
Ô
Ï
Ù
Ò
Þ Ý
Ò Ü
Ú
Ñ
The component integrals are usual real integrals and are well de• ned. In the last form appropriate limits must placed in the integrals. Some very straightforward rules of integration are given below: 1.
á
2.
à
3.
à
â
ä á
à
á
4.
ã
ã
ä
ì
í
ä
å
å
á
æ
æ
ç
é
è
ê
ã î
å
ä
å
ä
â
æ
å
æ
å
ç
á
ã
å
è
ä
è
à
à
á
å
æ
where å
ã á
ã ì
ç
ä
ä
å
æ
å
æ
ç
ç
å
å
is a complex constant. â
é
é à
à
á
ê á
ã î
ä
ä
å
å
æ
æ
ç
ç
å
å
ë
ë
ï
ï
ï
ç
æ
à
ð
ñ
ò
ó
ô
õ
ö
ô
ï
÷
ð
ñ
ø
ò
ó
ô
ó
ù
õ
õ
ô
ú
ó
ù
õ
ø
ö
ù
û
5. If countour ø
ò
ó
ô
for all
ø
õ
÷
ÿ
ý
þ
then ÿ
, where
ü
Example 4.4 Find integral of and also along st line path from to
Example 4.6 Show that from to
from to and from
along a straight line .
to
to
along a semicircu-
for
Example 4.5 . Find the integral from lar path in upper plane given by
is length of the
and ÿ
Cauchy-Goursat Theorem
17
Cauchy-Goursat Theorem Theorem 4.1 (Jordan Curve Theorem) Every simple and closed contour in complex plane splits the entire plane into two domains one of which is bounded. The bounded domain is called the interior of the countour and the other one is called the exterior of the contour. De• ne a sense direction for a contour. Theorem 4.2 Let be a simple closed contour with positive orientation and let be the interior of If and are continuous and have continuous partial derivatives and at all points on and , then
!
"
#
"
!
#
&
'
$
(
(
)
)
"
*
+
-
(
%
)
&
,
"
*
+
,
*
+
&
/
(
.
)
(
" !
*
+
0 #
)
)
"
*
+
1
,
,
*
Theorem 4.3 (Cauchy-Goursat Theorem) Let be analytic in a simply connected domain If is any simple closed contour in , then 2
%
3
$
%
' &
(
4
4
+
,
.
5
2
(
Example 4.7 loop is zero.
4
4
+
6
(
.
"
7
8
3
4
(
9
+
"
:
;
4
<
etc are entire functions so integral about any +
2
6
Theorem 4.4 Let and be two simple closed positively oriented contours such that lies entirely in the interior of If is an analytic function in a domain that contains and both and the region between them, then $
=
$
6
$
$
=
3
2
%
6
$
=
$
' &
> (
4
4
+
' &
,
? (
4
4
.
+
2
,
2
3
'
(
4
4
(
Example 4.8 . Find Choose a circular contour inside +
.
@
4
4
A
+
2
$
B
if ,
2
is any contour containing origin. $
3
'
4
Example 4.9
4
,
.
F
G
if H
=
B C
D
C
E
?
K
I
contains $
3
'
6 4
C
J
C
Example 4.10 Find where to multiply connected domains.
.
Extend the Cauchy Goursat theorem
F
6
B
$
L
M
M
3
C
Antiderivative Theorem 4.5 (Fundamental Theorem of Integration) Let be de• ned in a simply connected domain and is analytic in . If and are points in and is any contour in joining and then the function 4
I
4
2
4
I
4
%
%
%
"
%
(
N
4
&
+
'
(
.
4
4
+
2
,
$
18
Chapter 4 Integrals is analytic in O
and P
Q
R
S
T
U
V
R
S
T
De• nition 4.4 If is analytic in integral is de• ned as V
Z
W
and O
S
and X
S
are two points in Y
O
then the de• nite
[
\
V
Z
where P
R
S
T
^
S
U
P
R
S
Y
T
_
P
R
S
X
T
]
is an antiderivative of . V
c
Y
`
b
a
c
Y
Example 4.11
`
a
Y
X
X
Y
S
^
S
U
S
d
e
f
g
U
W
g
d
h
i
d
a
b
Example 4.12
a
S
U
S
o
U
W _
X
i
X
j
Z
k
l
l
m
n
l
m
n
l
m
n
p
[
Z
b
Example 4.13
Z
U
Z ]
r
S
Y
_
r
S
X
W
q
k
s
k
s
Cauchy Integral Formula Theorem 4.6 (Cauchy Integral Formula) Let be analytic in domain . Let positively oriented simple closed contour in . If is in the interior of then
be a
c
V
O
O
S
t
t
c
\
w
c
V
V
R
S
T
R
S
T
^
S
U
W u
v
p
S
S _
w
i
b
Example 4.14
Find
[
X
Z
V
R
S
T
U
W
`
x
V
R
S
T
^
if S
u
t
o y
S
o _
U
W
w
i
Z
b
Example 4.15
Z
V
Theorem 4.7 If at that point.
R
S
T
U
W
V
Y
`
X
Find V
R
S
T
^
if S
is square with vertices on t
u
R
z
{
u
z
T
is analytic at a point, then all its derivatives exist and are analytic c
\
w
c
V
V
|
}
~
R
S
T
U
u
R
S
T
^
S
v
`
}
R
i
S _
S
T
X
W
5
Series
Convergence of Sequences and Series Example 5.1
Example 5.2
Example 5.3
De• nition 5.1 An in• nite sequece if for each positive , there exists positive integer
of complex numbers has a such that
¡
¢
£
¤
¥
¦
§
¨
The sequences have only one limit. A sequence said to converge to if is its limit. A sequence diverges if it does not converge. ©
Example 5.4
converges to 0 if ª
©
ª
«
©
¥
Example 5.5 ©
ª
«
®
¯
±
²
³
¥
±
®
´
ª
«
µ
À
else diverges. ®
²
²
¶
±
and ª
ª
¿
converges to .
ª
·
if and only if
¬
¯
±
©
©
¥
°
Theorem 5.1 Suppose that
¬
¸
©
«
²
Then,
º
»
¼
½
¾
¼
Ê
Ê
Ï
Ð
Ñ
¨
¹
Ê
De• nition 5.2 If
¶
µ
Á
»
½
©
Ã
½
¾
Ã
Ä
Å
Æ
Ë
Ç
Ì
È
Î
Í
Í
É
Â
is a sequence, the in• nite sum
Ê
Ð
Ò
Ó
Ð
Ô
Ó
Õ
Õ
Õ
Ó
Ð
Ó
Õ
Õ
Õ
is called
Ì
Ö
a series and is denoted by
. ×
Ø
Ù
Ú
Û
Ý
De• nition 5.3 A series sums
is said to converge to a sum Ü
Ø
Ù
Ú
Û
Û
Þ
converges to . ç
ß
à
á
â
ã
á
ä
ã
å
å
å
ã
á
æ
Þ
if a sequence of partial
20
Chapter 5 Series Theorem 5.2 Suppose that è
é
ê
ë
é
ì
í
î
and é
ï
ê
ð
ñ
ò
ó
Then, ô
õ
ö
÷
÷
if and only if
ø
ù
û
ü
ú
ý
ý
ö
ö
÷
÷
ø
ù
û
ÿ
÷
þ
÷
ø
û
ù
ý
Example 5.6
if
÷
û
ú
ú
ú
Taylor Series Theorem 5.3 (Taylor Series) If is analytic in a circular disc of radius at then at each point inside the disc there is a series representation for
ý
andcentered given by
ú
ö
÷
û
÷
ú ÷
ú ø
ú
where
÷
÷
û
ú
ý
Example 5.7
÷
ù
û
ú
ú
ý
÷
ù
Example 5.8
÷
û
ù
ú
ú
ý
Example 5.9
÷
û
ú
ý
÷
÷
ù
Example 5.10
û
ú
ú
Laurent Series Theorem 5.4 (Laurent Series) If is analytic at all points in an annular region such that then at each point in there is a series representation for given by
ú
ù
ú
ú
ý
ý
ö
ö
÷ ÷
û
÷
÷
ú ÷
ú ø
ú ÷
ø
ù
where ú
#
ú
%
$
÷
û
÷
ù
ú
!
ú
"
#
%
$
ú
ú
÷
û
÷
ú
!
"
ú
ú
&
ú
ù
Laurent Series and
is any contour in '
21
. (
Example 5.11 If is analytic inside a disc of radius about then the Laurent series for is identical to the Taylor series for . That is all . )
*
)
)
Example 5.12
2
2
0
5
>
@
A
@
B
P
3
8
<
where =
@
A
@
B
C
/
0
,
-
1
D
9
4
K
E
;
?
7
2
Example 5.13 and
:
6
/
2
+
.
F
A
G
Find Laurent series for all
K
H
I
J
>
D >
I
J
L
I
M
L
D
Example 5.14 Note that Q
H
E J
J
M
R
S
T
U
F
A
G
V
A
D
@
A
@
N
C
O
C
N
@
A
@
N
P
6
Theory of Residues And Its Applications
Singularities De• nition 6.1 If a function fails to be analytic at but is analytic at some point in each neighbourhood of , then is a of . W
X
X
Y
X
Y
Z
[
\
]
^
_
`
a
b
c
[
\
Y
d
e
De• nition 6.2 If a function fails to be analyitc at for some then is said to be an e
h
i
j
j
f
k
f
i
l
l
e
Example 6.1
|
}
~
{
|
}
~
Example 6.2
{
|
}
~
{
|
}
~
Example 6.4
Z
c
_
g
`
d
n
o
but is analytic at each of . p
q
r
s
t
u
v
w
x
y
q
r
z
f
in
{
[
has an isolated singularity at . }
Example 6.3 , also has a singularity at {
f
m
g
}
|
}
|
has isolated singularities at ~
}
~
}
has isolated singularities at
}
.
for integral
.
all points of negative x-axis are singular. }
Types of singularities If a function points in part
{
}
¢
¡
in the prinipal part then
±
²
but for all then it is called a simple pole. ¨
£
¤
¨
¥
³
®
3. If all s are zero then ª
is called an essential singu-
¡
2. If for some integer order of If ®
1. If there are in• nite nonzero larity of
has an isolated singularity at then a such that is analytic at all . Then must have a Laurent series expansion about . The is called the principal part of
¯
°
¦
©
§
¨
ª
«
©
¬
®
is called a removable singularity.
then ¯
°
is called a pole of
24
Chapter 6 Theory of Residues And Its Applications Example 6.5 singularity at
´
µ
¶
¶
·
¸
¸
½
µ
¹
º
»
¶
·
¼
is unde• ned at ¶
¶
¸
½
¾
has a removable isolated ´
¾
Residues Suppose a function has an isolated singularity at that is analytic for all in deleted nbd representation ´
´
¶
½
Ã
Ä
¶
Å
¶
¿
¶
Ç
¿
Ä
then there exists a such . Then has a Laurent series À
Á
Ã
Á
Â
½
´
Ç
È È
Ë
È
È
´
µ
¶
·
¸
Æ
µ
¶
Å
¶
¿
·
Æ
¾
µ
È
É
È
¿
É
¶
¶ Å
Ì
· ¿
Í
Ê
The coef• cient Ì
¸
Î Ï
where
´
Ð
Ñ
Ò
µ
¶
·
Ô
¶
Ó
is any contour in the deleted nbd, is called the residue of
at
Í
Õ
´
¶
¿
¾
Ì
Example 6.6 wise . ´
µ
¶
·
. Then
¸ Ö
×
Ö
Ø
Ï
Ð
Ñ
Ï
Ð
if Ñ
Ó
Ì
Ù
´
µ
¶
·
Ô
¶
¸
¸
contains Õ
¶
¿
, other-
½
Í
Ì
Example 6.7 ´
µ
¶
·
Ö
Example 6.8 ó
ô
í
õ
ð
Ú
Ö
í
ö
×
Ö
÷
Ø
Û
ø
Ü
ù
Ý
ó
ô
í
õ
ð
ö
÷
ù ø
ð
ü
ß
à
á
â
ã
ä
â
å
æ
ç
è
if
ý
. Show
ú
û
í
ü
if
é
Þ
. Show
ú
û
Example 6.9 has a singularity at
Show
¸
þ
ý
ó
þ
ô
ó
í
ô
õ
í
ÿ
õ
í
ÿ
ð
í
ð
if
ê
ê
ë
ë
ì
ê
ë
ì
ì
í
î
í
í
ì
ï
ì
ð
ì
ñ
ð
ð
ñ
ò
ò
even though it ñ
ò
Theorem 6.1 If a function is analytic on and inside a positively oriented countour , except for a • nite number of points inside , then ó
ê
í
í
ò
ò
ò
í
ú
Res
Example 6.10 Show that
!
"
#
$
%
&
'
Residues of Poles Theorem 6.2 If a function
has a pole of order (
at )
then @
9
:
;
<
"
*
:
4
Res
6 4
4
(
5
9
,
#
-
.
>
=
=
:
*
=
6
+
"
;
<
/
0
1
2
3
7
A
A
3
Example 6.11 B
C
D
E
F
G
H
I
H
J
?
4
=
7
8
Simple pole Res B
C
D
K
E
F
L
7
7
A
Quotients of Analytic Functions Example 6.12 order at . Res M
^
O
Q
Example 6.13
N
O
P
_
M
N
O
P
Q
M
. Simple pole at .
R
S
N
_
T
S
P
U
Q
Q
e
V
W
`
a
d
S
b
f
[
c
d
X
\
h
i
Q
Y
Z
Res M
N
Y
P
Q
[
\
[
. Pole of ]
]
. Res d
g
[
O
25
k
l
m
n
o
p
q
r
s
t
. Res k
l
s
n
o
p
u
v
p
s
w
x
r
y
s
t
z
j
Quotients of Analytic Functions Theorem 6.3 If a function 1. 2.
is singular at
iff
has a simple pole at
k
l
if
{
n
o
|
}
~
, where
and
are analytic at
then
. Then residue of
at
is
.
A
References
This appendix contains the references.
B
Index
This appendix contains the index.
