Complex Analysis

1. Complex Analysis
PH 503 CourseTM
Charudatt Kadolkar
Indian Institute of Technology, Guwahati

2. ii
Copyright
¡
2000 by Charudatt Kadolkar

3. 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.

4. Contents
Preface iii
Preface Head iii
Acknowledgments iii
1 Complex Numbers 1
De• nitions 1
Algebraic Properties 1
Polar Coordinates and Euler Formula 2
Roots of Complex Numbers 3
Regions in Complex Plane 3
2 Functions of Complex Variables 5
Functions of a Complex Variable 5
Elementary Functions 5
Mappings 7
Mappings by Elementary Functions. 8
3 Analytic Functions 11
Limits 11
Continuity 12
Derivative 12
Cauchy-Riemann Equations 13

5. vi Contents
Analytic Functions 14
Harmonic Functions 14
4 Integrals 15
Contours 15
Contour Integral 16
Cauchy-Goursat Theorem 17
Antiderivative 17
Cauchy Integral Formula 18
5 Series 19
Convergence of Sequences and Series 19
Taylor Series 20
Laurent Series 20
6 Theory of Residues And Its Applications 23
Singularities 23
Types of singularities 23
Residues 24
Residues of Poles 24
Quotients of Analytic Functions 25
A References 27
B Index 29

6. 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:
§ ¨ © © ! # $ % ' # $ ' % (
! # ' # ( ) 0 1 2 3 4 2 5 6 0 1 7 1 2 8 9 @ 9 A B C @ 9 A D C A 9 @ E
In the rest of the chapter use F G F H G F I G P P P for complex numbers and Q R S for real numbers.
introduce T and U V W X T Y notation.
Algebraic Properties
1. Commutativity
U ` X U a V U a X U ` b U ` U a V U a U `
2. Associativity
c
U ` X U a d X U e f g h i p g q i g e r s p g h g q r g e f g h p g q g e r
3. Distributive Law
g p g h i g q r f g g h i g g q
4. Additive and Multiplicative Indentity
g i t u v w v x y € 
5. Additive and Multiplicative Inverse
‚
 € ƒ
‚ „ … ‚ † ‡
 ˆ ‰ 
‘ ’
“ ” • – ” — ˜
–
“ ” • – ” ™ d e f
g h

7. 2 Chapter 1 Complex Numbers
6. Subtraction and Division
i j k i l m i j n o k i l p q
i j
i l
m i j i r
j
l
7. Modulus or Absolute Value s t s u v w x y z x
8. Conjugates and properties { | w } ~ z |  w € } z 
{ ‚ ƒ „ … † „ ‡ ƒ „ …
„ ‡ „ … † „ ‡ „ …
ˆ
„
‡
„ … ‰
†
„
‡
„ …
9. Š
„
Š
…
† „ „
‹ Œ  Ž
  
 ‘ ’ “ ” •
” – ”
— ˜
10. Triangle Inequality ™
” š › ” œ
™  ž Ÿ ž ¡ ž Ÿ ¢ ž
Polar Coordinates and Euler Formula
1. Polar Form: for
Ÿ £
¤ ¥
, Ÿ
¤ ¦ § ¨ © ª « ¬ ­ ® ¯ ° ± ²
where ³ ´ µ ¶ µ and · ¸ ¹ º » ¼ ½ ¾ . ¿ is called the argument of À . Since ¿ Á Â Ã Ä is also
an argument of Å Æ the principle value of argument of Å is take such that Ç Ä È É Ê
Ë Ì
For Í Î Ï the Ð Ñ Ò Ó is unde• ned.
2. Euler formula: Symbollically,
Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ Û ß à Ü á
3. â ã â ä
× å
ã
å
ä æ ç è é ê ë ì í î
ï ð
ï ñ ò ó
ð
ó
ñ ô õ ö ÷ ø ù ÷ ú
î
ï û ü ý û þ û ÿ
4. de Moivre’s Formula ¡ ¢ £ ¤ ¥ ¦ £ § ¨ ¤ ©
û
ü
¡ ¢ £ ¤ ¥ ¦ £ § ¨ ¤

8. Roots of Complex Numbers 3
Roots of Complex Numbers
Let then
! # $ %
' (
) 0 1 2 34 5 5
There are only 4
distinct roots which can be given by
2 6 7 8 9 8 @ @ @ 8
4 A
9 @
If B is a
principle value of C D E F then B G
4
is called the principle root.
Example 1.1 The three possible roots of H I P Q
R S T U V W X Y ` a b c d e f g h
are i p q
g f r s
i p q
g f r t
p
r
q
g h s
i p q
g f r t
p u q
g h
.
Regions in Complex Plane
1. v w x y € of  ‚ is de• ned as a set of all points ƒ which satisfy
„
ƒ … ƒ
‚
„ † ‡
2. ˆ ‰  ‘ ’ “ ” • – — of ˜ ™ is a nbd of ˜ ™ excluding point ˜ ™ .
3. Interior Point, Exterior Point, Boundary Point, Open set and closed set.
4. Domain, Region, Bounded sets, Limit Points.

9. 2 Functions of Complex Vari-
ables
Functions of a Complex Variable
A d e f g h i j k l de• ned on a set m is a rule that uniquely associates to each point n of m a
a complex number o p Set q is called the r s t u v w of x and y is called the value of x at z
and is denoted by x { z | } y ~
x { z | } x {  €  ‚ | } ƒ „ … † ‡ ˆ ‰ Š ‹ Œ  Ž  
‘
Œ ’  “
‘
Œ ” • – —  “ ˜ Œ ” Ž ™  š › ‹ Œ ” Ž ™  “ œ  ž Ÿ ¡ ¢ £ ¤ ¥ ¦ § ¨ ©
Example 2.1 Write ª « ¬ ­ ® ¯ ° ¬ ± in ² ³ ´ µ form.
² « ¶ · ¸ ­ ® ¹ º » ¼ ½
¾ ¿
½ À
¼
½ Á
½
and Â Ã Ä Å Æ Ç È É
¿
¼
¾ ¿
½ À
¼
½ Á
½
Ê
Ã Ë Å Ì Ç È Ë Í
É Î Ï Ð Ñ
Ì and Â Ã Ë Å Ì Ç È Ò Ë Í
É Ð Ó Ô Ñ
Ì
Domain of Õ is Ö × Ø Ù Ú .
A Û Ü Ý Þ ß à á â ã ä å æ ç è é ê ç ë ì í î ï ë is a rule that assigns more than one value to each point of
domain.
Example 2.2 ð ñ ò ó ô õ ö ÷ This function assigns two distinct values to each ö ø ù
ú û ü
.
One can choose the function to be single-valued by specifying
õ
ö
ú ý
õ þ ÿ
¡ ¢ £
where ¤ is the principal value.
Elementary Functions

10. 6 Chapter 2 Functions of Complex Variables
1. Polynomials ¥ ¦ § ¨ ©
whrere the coef• cients are real. Rational Functions.
2. Exponential Function
! #
# $
% ' (
)
0 1 2 3 3 3
Converges for all 4 5 For real 4 the de• nition coincides with usual exponential func-
tion. Easy to see that 6 7 8 9 @ A B C
2 D
B E F C . Then
6 G 9 6 H I @ A B P
2 D
B E F P Q
a. 6
G R
6
G S
9 6
G R T G S
.
b. 6
G T U V
7 9 6
G
5
c. A line segment from I W X Y Q to I W X ` a Q maps to a circle of radius 6
H
centered at
origin.
d. No Zeros.
3. Trigonometric Functions
De• ne
@ A B 4 9
6 7
G
2
6 b 7
G
`
B E F 4 9
6 7
G c
6 b 7
G
`
d e
F 4 9
B E F 4
@ A B 4
a. B E F
U
4
2
@ A B
U
4 9 f
b. ` B E F 4 g @ A B 4
U
9 B E F I 4 g
2
4
U
Q
2
B E F I 4 g
c
4
U
Q
c. ` @ A B 4
g
@ A B 4
U
9 @ A B I 4
g
2
4
U
Q
2
@ A B I 4
g
c
4
U
Q
d. ` B E F 4 g B E F 4
U
9
c
@ A B I 4 g
2
4
U
Q
2
@ A B I 4 g
c
4
U
Q
e. B E F I 4
2
` a Q 9 B E F 4 and @ A B I 4
2
` a Q 9 @ A B 4 5
f. B E F 4 9 Y iff 4 9 h a I h 9 Y X i f X 5 5 5 Q
g. @ A B 4 9 Y iff 4 9
V
U
2
h a I h 9 Y X i f X 5 5 5 Q
h. These functions are not bounded.
i. A line segment from I Y X P Q to I ` a X P Q maps to an ellipse with semimajor axis
equal to @ A B p q under r s t function.
4. Hyperbolic Functions
De• ne
u v
r p w x y
€ 
y ‚
€
ƒ
r s t p w x
y
€ „
y ‚
€
ƒ

11. Mappings 7
a. … † ‡ ˆ ‰  ‘ ’ “  … † ‡ ‘ ” • – — ˜ ™ d e f g • – — e
b. • – — ˜ h e i — j k ˜ h e g l
c. — j k ˜ ™ e m n o d f g — j k ˜ ™ e f ” • – — ˜ ™ e m n o d f g • – — ˜ ™ e f
d. — j k ˜ e g p iff e g q o d ™ q g p r s l r t t t f
e. • – — ˜ e g p iff e g u v
h
m q o w d ™ q g p r
s
l r t t t f
5. Logarithmic Function
De• ne x y z { | x y z } ~  €  ~ ‚ ƒ „ …
then † ‡ ˆ ‰ Š ‹ Œ 
a. Is multiple-valued. Hence cannot be considered as inverse of exponetial func-
tion.
b. Priniciple value of log function is given by
Ž  
Œ ‹
Ž   ‘ ’ “ ”
where
”
is the principal value of argument of • .
c. – — ˜ ™ • š • › œ  – — ˜ • š ž – — ˜ • ›
Mappings
Ÿ
 ™ • œ . Graphical representation of images of sets under is called ¡ ¢ £ £ ¤ ¥ ¦ § Typi-
cally shown in following manner:
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 ³ plane. Use ³ © ´ µ ¨ ¶ ©
·
« ¬ ¸ © ¹ º » ¼ .
Example 2.3 ½ ¾ ¿ À .

12. 8 Chapter 2 Functions of Complex Variables
Mapping of Á Â
10.80.60.40.20-0.2-0.4-0.6
2
1.5
1
0.5
0
Mapping Á Â
1. A straight line Ã Ä Å maps to a parabola Æ Ç È É Ê Ë Ì Í Î Ï Ë Ì Ð
2. A straight line Ñ Ò Ë maps to a parabola Ó Ì Ò Ê Ë Ì Í Î Ï Ë Ì Ð
3. A half circle given by Ô Ò Õ Ö × Ø Ù where Ú Û Ü Û Ý maps to a full circle given by
Þ
Ò Õ
Ì
Ö
×
Ø Ì Ù ß
This also means that the upper half plane maps on to the entire complex
plane.
4. A hyperbola à Ì
Ï
Ñ Ì Ò á maps to a straight line â ã á ä
Mappings by Elementary Functions.
1. Translation by å æ is given by ç ã å è å æ ä
2. Rotation through an angle é æ is given by ç ã ê ë ì å ä
3. Rel• ection through x axis is given by ç ã í
î ï
4. Exponential Function

13. 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 ð ñ ò and ð ñ ó ô maps to the entire complex
plane.
5. Sine Function õ ö ÷ ø ñ õ ö ÷ ù ú û õ ü ð ý þ ú û õ ù õ ö ÷ ü 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 .

14. 3 Analytic Functions
Limits
A function ÿ is de• ned in a deleted nbd of ¡ ¢
De• nition 3.1 The limit of the function ÿ
£
¤ as ¥ ¦ § is a number ¨ § if, for any
given © there exists a such that
! # $ % ' ( ! ) 0 1
Example 3.1 # $ % 2 3 4 5 Show that 6 7 8 9 @ A B C D E F G H E I P
Example 3.2 C D E F G E Q P Show that R S T U V U B C D E F G E Q
I
P
Example 3.3 C D E F G E W XE P Show that the limit of C does not exist as E Y ` .
Theorem 3.1 Let C D E F G a D b c d F e f g D b c d F and h I G a I e f g I P R S T
U V U B
C D E F G h I if
and only if R S T
i p q r s
V
i p
B
q r
B
s
a G a I and R S T
i p q r s
V
i p
B
q r
B
s
g G g I P
Example 3.4 C D E F G t S u E . Show that the R S T
U
V
U B
C D E F G t S u E
I
Example 3.5 C D E F G v b e f d
Q
. Show that the R S T
U
V
Q w
C D E F G x f .
Theorem 3.2 If R S T
U
V
U B
C D E F G h
I
and R S T
U
V
U B y
D E F G €  ‚
ƒ „ …
† ‡ † ˆ ‰  ‘ ’ “ ” • ‘ ’ “ – — ˜ ™ d e ™ f
g h i
j k j l m n o p q n o p — ˜ ™ e ™ f
g h i
j k j l m n o p r q n o p —
˜ ™
e ™
e ™ s— t u
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:

15. 12 Chapter 3 Analytic Functions
1. v w x
y z y { | } | ~
2. v w x
y
z
y
{  € }  €

~
3. v w x
y z y
{ ‚ ƒ
 „ }
‚ ƒ


„
if ‚
is a polynomial in 
.
4. v w x
y
z
y
{ … † ‡
ƒ
 „ }
… † ‡
ƒ


„ ~
5. v w x
y
z
y { ˆ
w ‰
ƒ
 „ }
ˆ
w ‰


~
Continuity
De• nition 3.2 A function Š , de•ned in some nbd of 

is continuous at 

if
v w x
y
z
y {
Š
ƒ
 „ }
Š
ƒ


„ ~
This de• nition clearly assumes that the function is de• ned at 

and the limit on the LHS
exists. The function Š is continuous in a region if it is continuous at all points in that
region.
If funtions Š and ‹ are continuous at Œ  then Ž  ‹ , Ž ‹ and Ž  ‹ ‘ ‹ ‘ Œ  ’ “
” •
’ are also
continuous at Œ  .
If a function Ž ‘ Œ ’
” –
‘ — ˜ ™ ’  š › ‘ — ˜ ™ ’ is continuous at Œ

then the component functions
–
and › are also continuous at ‘ —  ˜ ™  ’ .
Derivative
De• nition 3.3 A function Ž , de•ned in some nbd of Œ  is differentiable at Œ  if
œ  ž
Ÿ Ÿ ¡
Ž ‘ Œ ’ ¢ Ž ‘ Œ  ’
Œ
¢
Œ 
exists. The limit is called the derivative of Ž at Œ  and is denoted by Ž £ ¤ ¥ ¦ § or ¨ ©
ª « ¬ ­ ® ¯
.
Example 3.6 °
¬ ­ ¯ ± ­ ² ³
Show that ° ´
¬ ­ ¯ ± µ ­ ³
Example 3.7 °
¬ ­ ¯ ± ¶ ­ ¶
²
. Show that this function is differentiable only at ­ ± ·
¸
. In
real analysis ¹ º ¹ is not differentiable but ¹ º ¹ » is.
If a function is differentiable at ¼ , then it is continuous at ¼ .

16. 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 ½ and ¾ be
differentiable at ¿ À
1. Á
Á Â
Ã
½ Ä ¾ Å
Ã
¿ Å Æ ½ Ç
Ã
¿ Å Ä ¾
Ã
¿ Å À
2. Á
Á Â
Ã
½ ¾ Å
Ã
¿ Å Æ ½ Ç
Ã
¿ Å ¾
Ã
¿ Å È ½
Ã
¿ Å ¾ Ç
Ã
¿ Å À
3. Á
Á Â
Ã
½ É ¾ Å
Ã
¿ Å Æ
Ã
½
Ç
Ã
¿ Å ¾
Ã
¿ Å Ê ½
Ã
¿ Å ¾
Ç
Ã
¿ Å Å É Ë ¾
Ã
¿ Å Ì Í if ¾
Ã
¿ Å ÎÆ Ï À
4. Á
Á Â
Ã
½ Ð Ñ Ò Ó Ô Ò Õ Ö × Ó Ñ Ó Ô Ò Ò Ñ × Ó Ô Ò Ø
5. Ù
Ù Ú Û
Õ Ü Ø
6. Ù
Ù Ú
Ô Ý Õ Þ Ô Ý ß à Ø
Cauchy-Riemann Equations
Theorem 3.3 If Ö
×
Ó Ô á Ò exists, then all the •rst order partial derivatives of component
function â Ó ã ä å Ò and æ Ó ã ä å Ò exist and satisfy Cauchy-Riemann Conditions:
â ç Õ æ è
â
è
Õ é æ
ç
Ø
Example 3.8 Ö Ó Ô Ò Õ Ô ê Õ ã ê
é
å ê ë ì í ã å Ø Show that Cauchy-Riemann Condtions
are satis•ed.
Example 3.9 Ö Ó Ô Ò Õ î Ô î
ê
Õ ã
ê
ë å
ê
Ø Show that the Cauchy-Riemann Condtions are
satis•ed only at Ô Õ Ü .
Theorem 3.4 Let Ö Ó Ô Ò Õ â Ó ã ä å Ò ë ì æ Ó ã ä å Ò be de•ned in some nbd of the point Ô á Ø If
the •rst partial derivatives of â and æ exist and are continuous at Ô
á
and satisfy Cauchy-
Riemann equations at Ô á , then Ö is differentiable at Ô á and
Ö
×
Ó Ô Ò Õ â
ç
ë ì æ
ç
Õ æ
è
é
ì â
è
Ø
Example 3.10 Ö Ó Ô Ò Õ ï ð ñ Ó Ô Ò Ø Show that Ö × Ó Ô Ò Õ ï ð ñ Ó Ô Ò Ø
Example 3.11 Ö Ó Ô Ò Õ ò ó ô Ó Ô Ò Ø Show that Ö × Ó Ô Ò Õ õ ö ò Ó Ô Ò Ø
Example 3.12 Ö Ó Ô Ò Õ ÷
ø ù
ø ú
Show that the CR conditions are satis•ed at û ü ý but the
function þ is not differentiable at ý
ú

17. 14 Chapter 3 Analytic Functions
If we write ÿ
¡ ¢ £ ¤
then we can write Cauchy-Riemann Conditions in polar coordi-
nates:
¥ ¦
§
¡ ¨
¤
¥
¤
©
¡
¨
¦
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 ÿ .
De• nition 3.6 A function is an entire function if it is analytic at all points of
Example 3.13 ÿ
§
ÿ is analytic at all nonzero points.
Example 3.14 ÿ ÿ is not analytic anywhere.
A function is not analytic at a point ÿ but is analytic at some point in each nbd of
ÿ then ÿ is called the singular point of the function .
Harmonic Functions
De• nition 3.7 A real valued function ! # $ % is said to be ' ( ) 0 1 2 3 in a domain of
xy plane if it has continuous partial derivatives of the •rst and second order and satis•es
4 5 6 7 8 9 @ @ A B C D E F G
: H I I P Q R S T U H V P Q R S T W X Y
Theorem 3.5 If a function `
P a T W b P Q R S T U c d P Q R S T
is analytic in a domain e then
the functions f and g are harmonic in e .
De• nition 3.8 If two given functions f h i p q r and g h i p q r are harmonic in domain D
and their •rst order partial derivatives satisfy Cauchy-Riemann Conditions
f s t g u
f
u
t v g
s w
then g is said to be x y €  ‚ ƒ „ … … ‚ ƒ † ‡ ˆ ‰  ‘ of ’ “
Example 3.15 Let ’ ” • – — ˜ ™ • d e — d and f ” • – — ˜ ™ g • — “ Show that f is hc of ’ and
not vice versa.
Example 3.16 ’ ” • – — ˜ ™ — h
e i
• — . Find harmonic conjugate of ’ “

18. 4 Integrals
Contours
Example 4.1 Represent a line segment joining points j k l m n and o p q r r n by parametric
equations.
Example 4.2 Show that a half circle in upper half plane with radius s and centered
at origin can be parametrized in various ways as given below:
1. t u v w x s y z { v | } u v w x s { ~  v | where v €  ‚ ƒ „
2. … † ‡ ˆ ‰ ‡ Š ‹ † ‡ ˆ ‰ Œ  Ž  ‡ Ž Š where ‡ €   ‚  „
3. … † ‡ ˆ ‰  †  ‡
 ‘
ˆ Š ‹ † ‡ ˆ ‰  
Œ
‡

‡ Ž Š where ‡ € 
‚ ‘
„
De• nition 4.1 A set of points ’ ‰ † … Š ‹ ˆ in complex plane is called an “ ” • if
… ‰ … † ‡ ˆ Š ‹ ‰ ‹ † ‡ ˆ Š † – — ‡ — ˜ ˆ
where … † ‡ ˆ and ‹ † ‡ ˆ are continuous functions of the real parameter ‡ „
Example 4.3 … † ‡ ˆ ‰ ™ š › ‡ Š ‹ † ‡ ˆ ‰ › œ  †  ‡ ˆ , where ‡ €  ‚  žƒ . Show that the
curve cuts itself and is closed.
An arc is called simple if ‡ Ÿ ‰ ‡
Ž ¡
’ † ‡ Ÿ ˆ ‰ ’ † ‡
Ž
ˆ „
An arc is closed if ’ † – ˆ ‰ ’ † ˜ ˆ .
An arc is differentiable if ’ ¢ † ‡ ˆ ‰ … ¢ † ‡ ˆ £ ¤ ‹ ¢ † ‡ ˆ exists and … ¢ † ‡ ˆ and ‹ ¢ † ‡ ˆ are continu-
ous. A smooth arc is differentiable and ’ ¢ † ‡ ˆ is nonzero for all ‡ .
De• nition 4.2 Length of a smooth arc is de•ned as
¥ ¦ § ¨ © ª «
¬ ­ ®
¯ ° ± ² ³ ´ µ ² ¶
The length is invariant under parametrization change.

19. 16 Chapter 4 Integrals
De• nition 4.3 A · ¸ ¹ º ¸ » ¼ is a constructed by joining •nite smooth curves end to end
such that ½ ¾ ¿ À is continuous and ½ Á ¾ ¿ À is piecewise continuous.
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 ½ ¾ ¿ À Ã Ä ¾ ¿ À Å Æ Ç ¾ ¿ À and a function
È
¾ ½ À Ã É ¾ Ä Ê Ç À Å Æ Ë ¾ Ä Ê Ç À is de• ned on it. The integral of
È
¾ ½ À along the contour  is
denoted and de• ned as follows:
Ì Í Î Ï Ð Ñ Ò Ð Ó Ô Õ
Ö
Î Ï Ð Ñ Ð × Ï Ø Ñ Ò Ø
Ó Ô
Õ
Ö
Ï Ù Ú
× Û Ü Ý ×
Ñ Ò Ø Þ ß Ô
Õ
Ö
Ï Ù
Ý ×
Þ
Ü
Ú
×
Ñ Ò Ø
Ó Ô Ï Ù Ò Ú Û
Ü
Ò
Ý
Ñ Þ ß Ô Ï Ù Ò
Ý
Þ
Ü
Ò 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. à á â ã ä å æ ç å è â à á ã ä å æ ç å where â is a complex constant.
2. à
á
ä ã ä å æ é ê ä å æ æ ç å è à
á
ã ä å æ ç å é à
á
ê ä å æ ç å ë
3. à á ì í á î
ã ä å æ ç å è
à á ì
ã ä å æ ç å é
à á î
ã ä å æ ç å ë
4. ï
ï ð ñ ò ó ô õ ö ô
ï
ï ÷ ð ñ ø ò ó ô ó ù õ õ ô ú ó ù õ ø ö ù û
5. If
ø
ò ó ô õ
ø ÷ ü
for all ý þ ÿ then
¡ ¢ £ ¤ ¥ ¦ § ¥
¨ ©
, where
is length of the
countour ÿ
Example 4.4
£ ¤ ¥ ¦ ¥
Find integral of
£
from
¤ ¦
to
¤ ¦
along a straight line
and also along st line path from
¤ ¦
to
¤ ¦
and from
¤ ¦
to
¤ ¦
.
Example 4.5
£ ¤ ¥ ¦ ¥
. Find the integral from
¤ ¦
to
¤ ¦
along a semicircu-
lar path in upper plane given by
¥
Example 4.6 Show that
¡ ¢
£ ¤ ¥ ¦ § ¥
¨ !
for
£ ¤ ¥ ¦ # ¥
$
and ÿ %
¥
from
to

20. 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 0 and 1 are continuous and have continuous partial derivatives
2 3 4 2 5 4
1
3
and 1
5
at all points on 6 and 7 , then
8 9 @
2
@ A
4 B C D
A E
1
@ A
4 B C D B C F
8 8 G
1
3
@ A
4 B C H 2
5
@ A
4 B C I D
A
D B
Theorem 4.3 (Cauchy-Goursat Theorem) Let P be analytic in a simply connected
domain 7 Q If 6 is any simple closed contour in 7 , then
8
9
P
@ R
C D
R
F S
Q
Example 4.7 P
@ R
C F
R T
4 U V W
@ R
C 4 X Y `
@ R
C
etc are entire functions so integral about any
loop is zero.
Theorem 4.4 Let 6 a and 6
T
be two simple closed positively oriented contours such
that 6
T
lies entirely in the interior of 6 a Q If P is an analytic function in a domain 7 that
contains 6 a and 6
T
both and the region between them, then
8
9 b
P
@ R
C D
R
F
8
9 c
P
@ R
C D
R
Q
Example 4.8 P
@ R
C F d e
R
. Find f
9
P
@ R
C D
R
if 6 is any contour containing origin.
Choose a circular contour inside 6 Q
Example 4.9 f
9
a
g h g i
D
R
F p q r
if 6 contains
R s
Q
Example 4.10 Find f
9
T
g t g
g
c u
T
where 6 v w
R
w
F p
Q Extend the Cauchy Goursat theorem
to multiply connected domains.
Antiderivative
Theorem 4.5 (Fundamental Theorem of Integration) Let P be de•ned in a simply
connected domain 7 and is analytic in 7 . If
R s
and
R
are points in 7 and 6 is any
contour in 7 joining
R s
and
R
4
then the function
x
@ R
C F
8 9
P
@ R
C D
R

21. 18 Chapter 4 Integrals
is analytic in y and €  ‚ ƒ „ … † ‚ ƒ „ ‡
De• nition 4.4 If † is analytic in y and ƒ ˆ and ƒ ‰ are two points in y then the de•nite
integral is de•ned as  ‘
’
 “
† ‚ ƒ „ ” ƒ … € ‚ ƒ ‰ „ • € ‚ ƒ ˆ „
where € is an antiderivative of † .
Example 4.11
‰ – —
˜ ™
ƒ
‰
” ƒ … ƒ d e f g
g
‰ – —
™
…
‰
d h i
ˆ ˆ
d
‡
Example 4.12
—
˜
ˆ j k l
ƒ …
l m n
ƒ o
—
ˆ
…
l m n
i
•
l m n p
‡
Example 4.13
 ‘
˜

“ q


… r
k s
ƒ ‰ • r
k s
ƒ ˆ ‡
Cauchy Integral Formula
Theorem 4.6 (Cauchy Integral Formula) Let † be analytic in domain y . Let t be a
positively oriented simple closed contour in y . If ƒ
™
is in the interior of t then
† ‚ ƒ
™
„ …
p
u v
i
’ w
† ‚ ƒ „ ” ƒ
ƒ
•
ƒ
™
‡
Example 4.14 † ‚ ƒ „ …
ˆ

‘
– x
‡ Find
˜
w
† ‚ ƒ „ ” ƒ if t y
o
ƒ •
i
o
…
u
‡
Example 4.15 † ‚ ƒ „ …

‰

– ˆ
‡ Find
˜
w
† ‚ ƒ „ ” ƒ if t is square with vertices on ‚ z
u {
z
u
„ ‡
Theorem 4.7 If † is analytic at a point, then all its derivatives exist and are analytic
at that point.
† | } ~ ‚ ƒ
™
„ …  €
u v
i
’ w
† ‚ ƒ „ ” ƒ
‚ ƒ
•
ƒ
™
„
}
– ˆ

22. 5 Series
Convergence of Sequences and Series
Example 5.1  ‚ ƒ „ … †
Example 5.2 
‚
ƒ ‡
‚
Example 5.3  ‚ ƒ ˆ ‰ Š ‹ Œ ˆ „ 
‚
… † Ž
De• nition 5.1 An in•nite sequece    
Ž
 ‘ ‘ ‘   ‚  ‘ ‘ ‘ of complex numbers has a ’ “ ” “ •
 if for each positive – , there exists positive integer — such that
˜ ™ š › ™ ˜ œ  ž Ÿ ¡ ¢ £ ¤ ¥ ¦ § ¨
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 © ª « ©
ª
converges to 0 if ¬ © ¬ ­ ® else diverges.
Example 5.5 ©
ª
« ® ¯ °
¥ ± ² ³ ¥ ±
® ´ ¯
¥
converges to
²
.
Theorem 5.1 Suppose that © ª « µ ª
± ² ¶
ª and © « µ
± ² ¶ ¨
Then,
· ¸ ¹
ª º » ¼ ½ ¾ ¼
if and only if ¿ À Á
½ Â
» Ã ½ ¾ Ã Ä Å Æ Ç È É
Ê Ë Ì Í
Ê Î
Í
De• nition 5.2 If Ï Ð
Ê Ñ
is a sequence, the in•nite sum Ð Ò Ó Ð Ô Ó Õ Õ Õ Ó Ð
Ê
Ó Õ Õ Õ is called
a series and is denoted by
Ì
Ö
× Ø Ù Ú Û
.
De• nition 5.3 A series Ü
Ý
Û
Ø Ù Ú Û
is said to converge to a sum Þ if a sequence of partial
sums
Þ ß à á â ã á ä ã å å å ã á æ
converges to ç .

23. 20 Chapter 5 Series
Theorem 5.2 Suppose that è é ê ë é ì í î é and ï ê ð ñ ò ó ô Then,
õ
ö
÷ ø ù ú
÷ û ü
if and only if ý
ö
÷ ø ù þ
÷ û ÿ ¡ ¢
ý
ö
÷ ø ù
£
÷ û ¤
Example 5.6 ¥
ý
¦
ú
÷
û § ¨ © §
ú
if
ú
§
Taylor Series
Theorem 5.3 (Taylor Series) If is analytic in a circular disc of radius
¦
andcentered
at
ú
¦
then at each point inside the disc there is a series representation for given by
©
ú
û
ý
ö
÷ ø
¦
÷ ©
ú
ú
¦
÷
where
÷ û
÷
©
ú
¦
!
Example 5.7 #
¡
ú
û
¥
ý
¦
ú
$
÷ % ù
¨ ©
'
§
! (
Example 5.8
ù
ù % )
û
¥
ý
¦ ©
§
÷
ú
÷
ú
§
(
Example 5.9 0
)
û
¥
ý
¦
ú
÷
¨
! (
Example 5.10
ù
)
û
¥
ý
¦ ©
§
÷
©
ú
§
÷
ú
§
§
(
Laurent Series
Theorem 5.4 (Laurent Series) If is analytic at all points
ú
in an annular region 1
such that
ù
ú
ú
¦
$
then at each point in 1 there is a series representation
for given by
©
ú
û
ý
ö
÷ ø
¦
÷
©
ú
ú
¦
÷
'
ý
ö
÷ ø ù
2
÷
©
ú
ú
¦
÷
where
÷ û
§
3 4
5 6
©
ú
7
ú
©
ú
ú
¦
÷ % ù
2
÷ û
§
3 4
5
6
©
ú
7
ú
©
ú
ú
¦
8
÷ % ù

24. Laurent Series 21
and 9 is any contour in @ .
Example 5.11 If A is analytic inside a disc of radius B about C D E then the Laurent
series for A is identical to the Taylor series for A . That is all F G H I .
Example 5.12 P
P Q R
H S T
G U
P V W
P X Y ` a
b c
where d e d f g h
Example 5.13 i p e q r s t
u
b
s t v
u
b
s w v
h Find Laurent series for all d e d x g y g x d e d x €
and d e d f € h
Example 5.14 Note that 
t
r
t
w ‚ ƒ „ …
i p e q † e h

25. 6 Theory of Residues And Its
Applications
Singularities
De• nition 6.1 If a function ‡ fails to be analytic at ˆ ‰ but is analytic at some point in
each neighbourhood of ˆ ‰ , then ˆ ‰ is a  ‘ ’ “ ” • – — ˜ ™ ‘ ’ d of e .
De• nition 6.2 If a function e fails to be analyitc at f g but is analytic at each f in
h i j
f k f g
j i l
for some
l m
then e is said to be an ‘  ™ • – d n o p q r s t u v w x y q r z of { .
Example 6.1 { | } ~  €  } has an isolated singularity at ‚ .
Example 6.2 { | } ~  €  ƒ „ … | † } ~ has isolated singularities at }  ‚ ‡ ˆ € ‡ ‰ ‰ ‰ .
Example 6.3 { | } ~  €  ƒ „ … | †  } ~ has isolated singularities at }  €  Š for integral
Š , also has a singularity at }  ‚ .
Example 6.4 { | } ~  ‹ Œ  } all points of negative x-axis are singular.
Types of singularities
If a function { has an isolated singularity at } Ž then  a  such that ‘ is analytic at all
points in ’ “ ” • – • — ” “  . Then ‘ must have a Laurent series expansion about • — . The
part ˜ ™š › œ 
œ ž
• – •
— Ÿ
š
is called the principal part of ‘ ¡
1. If there are in• nite nonzero  ¢ in the prinipal part then • — is called an essential singu-
larity of ‘ ¡
2. If for some integer £ ¤ ¥ ¦ §
¨ ©
but ª «
¨ ©
for all ¬ ­ ® then ¯ ° is called a pole of
order ® of ± ² If ®
¨ ³
then it is called a simple pole.
3. If all ª s are zero then ¯ ° is called a removable singularity.

26. 24 Chapter 6 Theory of Residues And Its Applications
Example 6.5 ´ µ ¶ · ¸ µ ¹ º » ¶ · ¼ ¶ is unde•ned at ¶ ¸ ½ ¾ ´ has a removable isolated
singularity at ¶ ¸ ½ ¾
Residues
Suppose a function ´ has an isolated singularity at ¶ ¿ À then there exists a Á Â ½ such
that ´ is analytic for all ¶ in deleted nbd ½ Ã Ä ¶ Å ¶
¿
Ä Ã Á . Then ´ has a Laurent series
representation
´ µ ¶ · ¸ Æ
Ç
È É
¿ Ê
È
µ ¶ Å ¶ ¿ ·
È Ë
Æ
Ç
È É Ì Í
È
µ ¶
Å
¶
¿
·
È
¾
The coef• cient
Í
Ì
¸ Î
Ï Ð Ñ Ò Ó
´ µ ¶ · Ô ¶
where Õ is any contour in the deleted nbd, is called the residue of ´ at ¶ ¿ ¾
Example 6.6 ´ µ ¶ · ¸
Ì
Ö × Ö Ø
. Then Ù
Ó
´ µ ¶ · Ô ¶ ¸
Ï Ð Ñ
Í
Ì
¸
Ï Ð Ñ
if Õ contains ¶ ¿ , other-
wise ½ .
Example 6.7 ´ µ ¶ · ¸
Ì
Ö Ú Ö × Ö Ø Û Ü Ý
Show Þ ß à á â ã ä â å æ ç è
é
if ê ë ì í î ï ì ð ñ ò
Example 6.8 ó ô í õ ð í ö ÷ ø ù ú
û ü
. Show ý þ ó ô í õ ÿ í ð
¡
if ê ë ì í ì ð ñ ò
Example 6.9 ó ô í õ ð ö ÷ ø
ù ú
û ¢
ü
. Show ý
þ ó ô í õ ÿ í ð £ if ê ë ì í ì ð ñ even though it
has a singularity at í ð £ ò
Theorem 6.1 If a function ó is analytic on and inside a positively oriented countour
ê , except for a •nite number of points í
ú ¤
í ¥ ¦ ò ò ò ¦ í § inside ¨ , then
©
§
! #
Res
!
$
Example 6.10 Show that % ' ( )
' 0 ' ( 1 2 3 4 5 6 7 8 9
Residues of Poles
Theorem 6.2 If a function @ has a pole of order A at
4 B
then
Res@ C
4 B D 5 E F G
' H ' I P Q
R S T
Q U V
W X Y `
W a
X Y `
R R
a
T
a b
U
X c
R
a
U U
d e
Example 6.11 f g h i p q
r s r t
e
Simple pole Res f g h u i p v
e

27. Quotients of Analytic Functions 25
Example 6.12 w x y €  ‚
ƒ „ ƒ … ƒ † ‡ ˆ
. Simple pole at y  ‰  Res w x ‰ €  ‘ ’ ‘ “ . Pole of
order ” at y  • . Res w x • €  – ‘ ’ ‘ “ .
Example 6.13 w x y €  — ˜ ™ d
d e f d g h i j
. Res 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
Quotients of Analytic Functions
Theorem 6.3 If a function k l { n o | } ~ 
€  ‚ ƒ
, where „ and … are analytic at † ‡ ˆ then
1. ‰ is singular at † ‡ iff … Š † ‡ ‹ Œ  Ž
2. ‰ has a simple pole at † ‡ if …  Š † ‡ ‹ Œ  . Then residue of ‰ at † ‡ is „ Š † ‡ ‹ ‘ …  Š † ‡ ‹ .

28. A References
This appendix contains the references.

29. B Index
This appendix contains the index.