Complex Analysis PH 503 Coursetm

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Complex Analysis PH 503 CourseTM

Charudatt Kadolkar Indian Institute of Technology, Guwahati

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 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 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 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 2 2 3 4 5

In the rest of the chapter use . for complex numbers and for real numbers.

7 8 9 : 6 ; introduce 6 and notation.

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 C D F C G C A H

3. Distributive Law

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 T U R V W

Q P S

[ \

g h

Q X Y Z

] ^ _ ` ^ a ] ^ b _ ` ^ c d e f 2 Chapter 1 Complex Numbers

6. Subtraction and Division

i j

i j k i l m i j n o k i l p q m i j i r

i l

t s u v w x y z x

7. Modulus or Absolute Value s

| w } ~ z | w } z

8. Conjugates and properties {

{

¡ ¢

10. Triangle Inequality

Polar Coordinates and Euler Formula

¤ ¥

1. Polar Form: for ,

¤ ¦ § ¨ © ª « ¬ ® ¯ ° ± ²

´ µ ¶ µ · ¸ ¹ º » ¼ ½ ¾ ¿ À ¿ Á Â Ã Ä

where ³ and . is called the argument of . Since is also

Æ Å Ç Ä È É Ê

an argument of Å the principle value of argument of is take such that

Ë Ì

Î Ï Ð Ñ Ò Ó For Í the is unde•ned.

2. Euler formula: Symbollically,

Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ Û ß à Ü á

ã â ä ã ä æ ç è é ê ë ì í î

3. â

× å å

ï ð ð

ï ñ ò ó ñ ô õ ö ÷ ø ù ÷ ú

ï û ü ý û þ û ÿ

¡ ¢ £ ¤ ¥ ¦ £ § ¨ ¤ © ¡ ¢ £ ¤ ¥ ¦ £ § ¨ ¤

4. de Moivre’s Formula

û ü Roots of Complex Numbers 3

Roots of Complex Numbers

Let then

" ! # #

" " )

There are only distinct roots which can be given by If * is a

$ % & ' & ( ( ( & ' (

, - . * /

principle value of + then is called the principle root.

4 5 6 7 8 9 : ; < = > ? @ A B C D

0 1 2 3 E F G E F G F G E F G F K G

B H I C B H J H C D I C B H J C D Example 1.1 The three possible roots of are C .

Regions in Complex Plane

M N O P Q R S

1. L of is de•ned as a set of all points which satisfy

T T V W

S U S

Y Z [ \ ] ^ _ ` a b c b c b c 2. X 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. 2 Functions of Complex Vari- ables

Functions of a Complex Variable

e f g h i j k l m n m

A d de•ned on a set is a rule that uniquely associates to each point of a

p q r s t u v w x y x z

a complex number o Set is called the of and is called the value of at

{ z | } y ~

and is denoted by x

x { z | } x { | }

¡ ¢ £ ¤ ¥ ¦ § ¨ ©

« ¬ ® ¯ ° ¬ ± ² ³ ´ µ

Example 2.1 Write ª in form.

¾ ¿ ¾ ¿

« ¶ · ¸ ® ¹ º » ¼ ½ Â Ã Ä Å Æ Ç È É ¼

² and

¼ ¼

½ ½

½ À ½ Á ½ À ½ Á

Ã Ë Å Ì Ç È Ë Í Ì Â Ã Ë Å Ì Ç È Ò Ë Í Ì

Î Ï Ð Ñ É Ð Ó Ô Ñ

É and

Ö × Ø Ù Ú

Domain of Õ is .

Ü Ý Þ ß à á â ã ä å æ ç è é ê ç ë ì í î ï ë A Û is a rule that assigns more than one value to each point of

domain.

ú û ü

ñ ò ó ô õ ö ÷ ö ø ù Example 2.2 ð This function assigns two distinct values to each .

One can choose the function to be single- valued by specifying

¡ ¢ £

ú ý

õ õ þ ÿ ö

where ¤ is the principal value.

Elementary Functions

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 the de•nition coincides with usual exponential func-

$ % & ' ( ) * + * - . + ,

tion. Easy to see that . Then

$ / ' $ 0 1 ( ) * 2 * - . 2 3

$ $ ' $

4 / 5 / 4 6 / 5

a. / .

$ % ' $ #

6 7 8 /

b. /

1 9 : ; 3 1 9 : < = 3 $ c. A line segment from to maps to a circle of radius 0 centered at origin. d. No Zeros.

3. Trigonometric Functions

De•ne

$ % $ > %

/ /

( ) * " '

$ % $ > %

/ ? /

* - . " '

* - . "

@ A

. " '

( ) * "

* - . " ( ) * " ' B

a. 7

< * - . " C ( ) * " ' * - . 1 " C " 3 * - . 1 " C " 3

b. ?

7 7 7

< ( ) * " ( ) * " ' ( ) * 1 " " 3 ( ) * 1 " " 3

C C C

c. ?

7 7 7

< * - . " C * - . " ' ( ) * 1 " C " 3 ( ) * 1 " C " 3

d. ?

7 7 7

* - . 1 " < = 3 ' * - . " ( ) * 1 " < = 3 ' ( ) * " #

e. and

- . " ' ; " ' D = 1 D ' ; : E B : # # # 3

f. * iff

( ) * " ' ; " ' D = 1 D ' ; : E B : # # # 3 8 g. iff

h. These functions ar7 e not bounded.

; : 2 3 1 < = : 2 3

i. A line segment from 1 to maps to an ellipse with semimajor axis

) * F G H I J equal to ( under function.

4. Hyperbolic Functions

De•ne

P Q P

K L

H F M N O O R

P T P

H I J F M N

O O R S

Mappings 7

V W X Y Z [ \ ] Z U V W [ ^ _ ` a b c d e f g _ ` a e

a. U

` a b h e i a j k b h e g l

b. _

j k b c e m n o d f g a j k b c e f ^ _ ` a b c e m n o d f g _ ` a b c e f

c. a

j k b e g p e g q o d c q g p r s l r t t t f

d. a iff

` a b e g p e g u v m q o w d c q g p r l r t t t f

e. _ iff h

5. Logarithmic Function

y z { | x y z } ~ ~

De•ne x

then

a. Is multiple- valued. Hence cannot be considered as inverse of exponetial function.

b. Priniciple value of log function is given by

where is the principal value of argument of .

Mappings

¡ ¢ £ £ ¤ ¥ ¦ § . 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

¨ © ª « ¬ © ® ¯ ° ± ²

we call ¨ plane. Use

³ © ´ µ ¨ ¶ ©

2. Show its images on another complex plane, which we call ³ plane. Use

¬ ¸ © ¹ º » ¼

« .

¾ ¿ À Example 2.3 ½ . 8 Chapter 2 Functions of Complex Variables

1.5

0.5

-0.6-0.4-0.20 0 0.2 0.4 0.6 0.8 1

Á Â Â

Mapping of Á Mapping

Ä Å Æ Ç È É Ê Ë Ì Í Î Ï Ë Ì Ð

1. A straight line Ã maps to a parabola

Ò Ë Ó Ì Ò Ê Ë Ì Í Î Ï Ë Ì Ð

2. A straight line Ñ maps to a parabola

Ò Õ Ö × Ø Ù Ú Û Ü Û Ý

3. A half circle given by Ô where maps to a full circle given by

Ò Õ ×

Ì Ø Ì Ù ß

Ö This also means that the upper half plane maps on to the entire complex

plane.

à Ì Ñ Ì Ò á â ã á ä 4. A hyperbola Ï maps to a straight line

Mappings by Elementary Functions.

æ ç ã å è å æ ä

1. Translation by å is given by

æ ç ã ê ë ì å ä

2. Rotation through an angle é is given by

î ï

ã í 3. Rel•ection through x axis is given by ç

4. Exponential Function Mappings by Elementary Functions. 9

Exponential Function A vertical line maps to a circle.

A horizontal line maps to a radial line.

ñ ò ð ñ ó ô A horizontal strip enclosed between ð and maps to the entire complex

plane.

ö ÷ ø ñ õ ö ÷ ù ú û õ ü ð ý þ ú û õ ù õ ö ÷ ü ð 5. Sine Function õ

A vertical line maps to a branch of a hyperbola. ô A horizontal line maps to an ellipse and has a period of ó . 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 2

Example 3.1 Show that

+ , - . / - 3 2 4 5 6 7 8 7 * + , - . / - 3 2

Example 3.2 Show that 1

, - . / - 9 - : 2 + - ; <

Example 3.3 + Show that the limit of does not exist as .

, - . / = , > ? @ . A B C , > ? @ . D 1 / = 1 A B C 1 2 4 5 6 + , - . / D 1

Theorem 3.1 Let + and if

7 8 7 *

4 5 6 = / = 1 4 5 6 C / C 1 2

F G H I E F G H I E F G H I E F G H I

and only if E and

8 * * 8 * *

+ , - . / J 5 K - 4 5 6 8 + , - . / J 5 K -

7 * 1

Example 3.4 . Show that the 7

+ , - . / L > A B @ 4 5 6 8 + , - . / N B 7

Example 3.5 3 . Show that the .

3 M

4 5 6 + , - . / D 4 5 6 , - . / P Q R

Theorem 3.2 If 1 and

8 8

7 7 * 7 7 * O

S T U

V W V X Y Z [ \ ] ^ _ [ \ ] ` a b c d e c f

g h i

j k j l m n o p q n o p a b c e c f

g h i

b c

j k j l m n o p r q n o p a e c a s t u

e 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

w x

1. v

y z y { | } | ~

w x

2. v

z { } ~

y y

w x

3. v if is a polynomial in .

y z y }

{

w x

4. v

z {

y y } ~

w x w w

5. v

z } ~

y y {

Continuity

De•nition 3.2 A function , de•ned in some nbd of is continuous at if

v w x

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 are continuous at then , and are also

continuous at .

If a function is continuous at then the component functions

and are also continuous at .

Derivative

De•nition 3.3 A function , de•ned in some nbd of is differentiable at if

ª « ¬ ® ¯

£ ¤ ¥ ¦ § ¨ ©

exists. The limit is called the derivative of at and is denoted by or .

¬ ¯ ± ² ³ ¬ ¯ ± µ ³

° ´

Example 3.6 ° Show that

¬ ¯ ± ¶ ¶ ± ·

Example 3.7 ° . Show that this function is differentiable only at . In

º ¹ ¹ º ¹ »

real analysis ¹ is not differentiable but is. ¼ If a function is differentiable at ¼ , then it is continuous at . 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. ¾

Some rules for obtaining the derivatives of functions are listed here. Let ½ and be À

differentiable at ¿

Ã Ã Ã Ã

½ Ä ¾ Å ¿ Å Æ ½ Ç ¿ Å Ä ¾ ¿ Å À

1. Á

Á Â

Ã Ã Ã Ã Ã Ã

¾ Å ¿ Å Æ ½ Ç ¿ Å ¾ ¿ Å È ½ ¿ Å ¾ Ç ¿ Å À

2. ½

Á Â

Ã Ã Ã Ã Ã Ã Ã Ã Ã

Á ½ É ¾ Å ¿ Å Æ ½ ¿ Å ¾ ¿ Å Ê ½ ¿ Å ¾ ¿ Å Å É Ë ¾ ¿ Å Ì Í ¾ ¿ Å Æ Î Ï À Ç

3. Ç if

Á Â

½ Ð Ñ Ò Ó Ô Ò Õ Ö × Ó Ñ Ó Ô Ò Ò Ñ × Ó Ô Ò Ø

4. Á

Á Â

Õ Ü Ø

5. Ù

Ù Ú Û

Ý Õ Þ Ô Ý ß à Ø

6. Ô

Ù Ú

Cauchy- Riemann Equations

Ó Ô á Ò

Theorem 3.3 If Ö exists, then all the •rst order partial derivatives of component

Ó ã ä å Ò æ Ó ã ä å Ò

function â and exist and satisfy Cauchy- Riemann Conditions:

â ç Õ æ è

â Õ é æ Ø

è ç

Ó Ô Ò Õ Ô ê Õ ã ê å ê ë ì í ã å Ø Example 3.8 Ö Show that Cauchy- Riemann Condtions

are satis•ed.

Ö Ó Ô Ò Õ î Ô î Õ ã ë å Ø ê

Example 3.9 ê Show that the Cauchy- Riemann Condtions are

Õ Ü

satis•ed only at Ô .

Ó Ô Ò Õ â Ó ã ä å Ò ë ì æ Ó ã ä å Ò Ô á Ø

Theorem 3.4 Let Ö be de•ned in some nbd of the point If

â æ Ô

the •rst partial derivatives of and exist and are continuous at á and satisfy Cauchy-

á Ö Ô á

Riemann equations at Ô , then is differentiable at and

Ö Ó Ô Ò Õ â ë ì æ Õ æ ì â Ø

ç ç è è

Ó Ô Ò Õ ï ð ñ Ó Ô Ò Ø Ö × Ó Ô Ò Õ ï ð ñ Ó Ô Ò Ø

Example 3.10 Ö Show that

Ó Ô Ò Õ ò ó ô Ó Ô Ò Ø Ö × Ó Ô Ò Õ õ ö ò Ó Ô Ò Ø

Example 3.11 Ö Show that

ø ù

Ö Ó Ô Ò Õ ÷ û ü ý ú

Example 3.12 ø Show that the CR conditions are satis•ed at but the

þ ý function is not differentiable at ú

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 .

De•nition 3.6 A function is an entire function if it is analytic at all points of

ÿ ÿ

Example 3.13 is analytic at all nonzero points.

ÿ ÿ

Example 3.14 is not analytic anywhere.

A function is not analytic at a point ÿ but is analytic at some point in each nbd of

ÿ ÿ then is called the singular point of the function .

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

" # $ % & ' ( ( ) * + , - . /

1 1 2 3 4 5 6 7 0 8 2 3 4 5 6 9 : ;

: 0

2 = 6 9 > 2 3 4 5 6 7 ? @ 2 3 4 5 6 A

Theorem 3.5 If a function < is analytic in a domain then

C A

the functions B and are harmonic in .

D E F G H C D E F G H De•nition 3.8 If two given functions B and are harmonic in domain D

and their •rst order partial derivatives satisfy Cauchy- Riemann Conditions

B I J C K

K I M

B J L C

N O P Q R S T U U R S V W X Y Z [ \ ]

then C is said to be of

^ _ ` a b c _ d e a d f ^ _ ` a b c g _ a ] f \ Example 3.15 Let \ and Show that is hc of and

not vice versa.

e i

^ _ ` a b c a h _ a \ ] Example 3.16 \ . Find harmonic conjugate of 4 Integrals

Contours

k l m n o p q r r n Example 4.1 Represent a line segment joining points j and 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:

u v w x s y z { v | } u v w x s { ~ v | v

1. t where

2. where

3. where

De•nition 4.1 A set of points in complex plane is called an if

where and are continuous functions of the real parameter

Example 4.3 , where . Show that the

curve cuts itself and is closed.

An arc is called simple if

An arc is closed if .

¢ ¢ £ ¤ ¢ ¢ ¢

An arc is differentiable if exists and and are continu-

¢ ous. A smooth arc is differentiable and 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.

16 Chapter 4 Integrals

¸ ¹ º ¸ » ¼

De•nition 4.3 A · is a constructed by joining •nite smooth curves end to end

¾ ¿ À ½ Á ¾ ¿ À such that ½ is continuous and is piecewise continuous.

A closed simple contour has only •rst and last point same and does not cross itself.

Contour Integral

½ ¾ ¿ À Ã Ä ¾ ¿ À Å Æ Ç ¾ ¿ À

If Â is a contour in complex plane de•ned by and a function

È È

½ À Ã É ¾ Ä Ê Ç À Å Æ Ë ¾ Ä Ê Ç À ¾ ½ À Â ¾ is de•ned on it. The integral of along the contour is

denoted and de•ned as follows:

Ì Í Î Ï Ð Ñ Ò Ð Ó Ô Õ Î Ï Ð Ñ Ð × Ï Ø Ñ Ò Ø

Ó Ô Ï Ù Ú Ñ Ò Ø Þ ß Ô Ï Ù Þ Ú Ñ Ò Ø

Õ × Û Ü Ý × Õ Ý × Ü ×

Ö Ö

Ó Ô Ï Ù Ò Ú Û Ò Ñ Þ ß Ô Ï Ù Ò Þ Ò Ú Ñ

Ü Ý Ý Ü

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:

á â ã ä å æ ç å è â à á ã ä å æ ç å â

1. à where is a complex constant.

à ä ã ä å æ é ê ä å æ æ ç å è à ã ä å æ ç å é à ê ä å æ ç å ë

á á

2. á

ã ä å æ ç å è ã ä å æ ç å é ã ä å æ ç å ë

á ì í á î à á ì à á î

3. à ï

4. ï

ï 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

¡ ¢ £ ¤ ¥ ¦ § ¥

þ ÿ

5. If for all ý then , where is length of the

ø ø ÷ ü ¨ ©

ò ó ô õ

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

from to 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 2 be analytic in a simply connected

3 $ %

domain % If is any simple closed contour in , then

& ( 4 4

+ , . 5

2 3

( 4 4 6 ( 4 ( 4

+ . " 7 8 9 + " : ; < +

Example 4.7 2 etc are entire functions so integral about any

loop is zero.

= $

Theorem 4.4 Let $ and be two simple closed positively oriented contours such

$ = 3 2 %

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

' > ' ?

& ( 4 4 & ( 4 4

+ , . + ,

2 2 3

( 4 4 ( 4 4

+ . @ A + ,

B 2 $

Example 4.8 2 . Find if is any contour containing origin. 3

Choose a circular contour inside $

4 4 I

, . F G H

B $ 3

D C E

Example 4.9 C if contains ? K

C J C

. F

B $ L M M 3 Example 4.10 Find C where Extend the Cauchy Goursat theorem to multiply connected domains.

Antiderivative

4 I 4

Theorem 4.5 (Fundamental Theorem of Integration) Let 2 be de•ned in a simply

4 I 4

% % $

connected domain % and is analytic in . If and are points in and is any "

contour in % joining and then the function

( 4 & ' ( 4 4

N + . + , 2

18 Chapter 4 Integrals

P Q R S T U V R S T W

is analytic in O and

O S X S Y O

De•nition 4.4 If V is analytic in and and are two points in then the de•nite [

integral is de•ned as Z

V R S T ^ S U P R S Y T _ P R S X T

Z ] V

where P is an antiderivative of .

Y ` a

b c

Y ` a

Y X X

^ S U S d e f g U W

Example 4.11 S

d h i d

S U S o U W _

Example 4.12 X

X j k l l m n l m n l m n p

Z [

U r S Y _ r S X W

q Example 4.13 ]

k s k s

Cauchy Integral Formula

O t

Theorem 4.6 (Cauchy Integral Formula) Let V be analytic in domain . Let be a

S t

positively oriented simple closed contour in O . If is in the interior of then

\ w

V R S T ^ S

V R S T U W

u v

S S

[ u

o o

V R S T U W V R S T ^ S t y S _ U W x

Example 4.14 ` Find if

u { u

V R S T U W V R S T ^ S t R z z T W

` X

Example 4.15 Y Find if is square with vertices on

Theorem 4.7 If V is analytic at a point, then all its derivatives exist and are analytic

at that point. c

\ w

V R S T ^ S

V | } ~ R S T U

u v

` X

}

R S S T

_ i 5 Series

Convergence of Sequences and Series

Example 5.1

Example 5.2

Example 5.3

De•nition 5.1 An in•nite sequece of complex numbers has a

if for each positive , there exists positive integer such that

sequence diverges if it does not converge.

ª « © ¬ © ¬ ®

¥ ± ² ³ ¥ ± ¥ ²

Example 5.5 ª converges to .

± ² ¶ ± ² ¶ ¨

· ¸ ¹

ª º » ¼ ½ ¾ ¼

À Á

if and only if ¿

Ê Î

Ê Ë Ì Í Í

» Ã ½ ¾ Ã Ä Å Æ Ç È É

½ Â

Ê Ñ Ê

Ð Ð Ò Ó Ð Ô Ó Õ Õ Õ Ó Ð Ó Õ Õ Õ

De•nition 5.2 If Ï is a sequence, the in•nite sum is called Ì

a series and is denoted by Ö .

× Ø Ù Ú Û

Ý Þ

De•nition 5.3 A series Ü is said to converge to a sum if a sequence of partial

Ø Ù Ú Û

sums Û

Þ ß à á â ã á ä ã å å å ã á æ

converges to ç .

20 Chapter 5 Series

é ê ë é ì í î é ï ê ð ñ ò ó ô

Theorem 5.2 Suppose that è and Then,

÷ û ü

÷ ø ù ú ý

if and only if ý

ö ö

÷ û ÿ ¡ ¢ ÷ û ¤

÷ ø ù þ ÷ ø ù

¦ û § ¨ © § §

Example 5.6 ¥ if

ú ú ú

Taylor Series

Theorem 5.3 (Taylor Series) If is analytic in a circular disc of radius andcentered

at then at each point inside the disc thý ere is a series representation for given by

ú ú ú

÷ ø

where

÷ û

÷ ù

¡ û ¦ ¨ © §

Example 5.7

ú ú

û ¦ © § §

Example 5.8 ù

ú ú

û ¦ ¨ ¥

Example 5.9

÷ ÷

Example 5.10

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

ú ú

ý ý

for given by

ö ö

ú ú ú

÷ ø ÷ ø ù ú

where ú

# $

§ ©

÷ û

÷ ù

ú ú

! " ¦

ú ú

§ ©

÷ û

÷ ù

ú ú

! " ¦

ú ú

Laurent Series 21 (

and ' is any contour in .

* + , -

Example 5.11 If ) is analytic inside a disc of radius about then the Laurent

) . / 0 1

series for ) is identical to the Taylor series for . That is all .

2 : ; < =

2 0 5 6 @ A @ B C D

> ? 7

Example 5.12 / where

2 8 9

2 3 4

K K

E F A G H I J D @ A @ N C O C N @ A @ N P >

Example 5.13 > Find Laurent series for all

I J L I M L

A @ B P D

and @

Q H E F A G V A D

Example 5.14 Note that J

M R S T U 6 Theory of Residues And Its Applications

Singularities

X Y

De•nition 6.1 If a function W fails to be analytic at but is analytic at some point in

Y X Y Z [ \ ] ^ _ ` a b c [ \ d e

each neighbourhood of X , then is a of .

f g f

De•nition 6.2 If a function e fails to be analyitc at but is analytic at each in

h i j j i l l m

k f g e [ Z c _ ` d n o p q r s t u v w x y q r z {

f for some then is said to be an of .

| } ~ }

Example 6.1 { has an isolated singularity at .

| } ~ | } ~ }

Example 6.2 { has isolated singularities at .

| } ~ | } ~ }

Example 6.3 { has isolated singularities at for integral

}

, also has a singularity at .

| } ~ } Example 6.4 { all points of negative x- axis are singular.

Types of singularities

}

If a function { has an isolated singularity at then a such that is analytic at all

points in . Then must have a Laurent series expansion about . The

part 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

¨ © ¨ ©

¤ ¥ ¦ § ª « ¬ ® ¯ °

2. If for some integer £ but for all then is called a pole of

¨ ³

± ² ®

order ® of If then it is called a simple pole.

¯ ° 3. If all ª s are zero then is called a removable singularity.

24 Chapter 6 Theory of Residues And Its Applications

µ ¶ · ¸ µ ¹ º » ¶ · ¼ ¶ ¶ ¸ ½ ¾ ´

Example 6.5 ´ is unde•ned at has a removable isolated

¸ ½ ¾ singularity at ¶

Residues

¶ ¿ À Á Â ½

Suppose a function ´ has an isolated singularity at then there exists a such

¶ ½ Ã Ä ¶ Å ¶ Ä Ã Á ´ that ´ is analytic for all in deleted nbd . Then has a Laurent series

representation

Ç Ç

È Ë

´ µ ¶ · ¸ Æ µ ¶ Å ¶ ¿ · Æ ¾

µ ¶ ¶ ·

Å ¿

È É È É Ì Í

¿ Ê

The coef•cient

¸ Î ´ µ ¶ · Ô ¶

Ï Ð Ñ Ò Ó

´ ¶ ¿ ¾

where Õ is any contour in the deleted nbd, is called the residue of at

Ï Ð Ñ Ï Ð Ñ

´ µ ¶ · ¸ Ù ´ µ ¶ · Ô ¶ ¸ ¸ Õ ¶ ¿

× Ö Ø Example 6.6 Ö . Then if contains , other-

wise ½ .

´ µ ¶ · ¸ Þ ß à á â ã ä â å æ ç è ê ë ì í î ï ì ð ñ ò

Ú Ö × Ö Ø Û Ü Ý

Example 6.7 Ö Show if

ó ô í õ ð í ö ÷ ø ù ú ý þ ó ô í õ ÿ í ð ê ë ì í ì ð ñ ò ü

Example 6.8 û . Show if

ù ú ý

ó ô í õ ð ö ÷ ø þ ó ô í õ ÿ í ð £ ê ë ì í ì ð ñ

ü ¢

Example 6.9 û . Show if even though it

ð £ ò has a singularity at í

Theorem 6.1 If a function ó is analytic on and inside a positively oriented countour

í í ¥ ¦ ò ò ò ¦ í § ¨

ê , except for a •nite number of points inside , then

ú ¤

Res

Example 6.10 Show that

! " # $ % & '

Residues of Poles )

Theorem 6.2 If a function ( has a pole of order at then

@ A

9 : ; <

" *

: ?

4 4 4

= = > =

4 5 6

( + 9 =

; <

Res :

" * , # - . /

0 1 2 3

7 7 7

A A

3 7 8

B C D E F G B C D K E F L

I H J Example 6.11 H Simple pole Res

Quotients of Analytic Functions 25

M N O P Q R O Q Y Z M N Y P Q [ \ [ ]

T S U S V W X

Example 6.12 S . Simple pole at Res . Pole of

O Q _ M N _ P Q ` [ \ [ ]

order ^ at . Res .

N O P Q a b c d k l m n o p q r s t k l s n o p u v p s w x r y s t z

Example 6.13 M . Res . Res

d e f d g h i j

Quotients of Analytic Functions

l { n o | } ~

Theorem 6.3 If a function k , where and are analytic at then

1. is singular at iff

2. has a simple pole at if . Then residue of at is . A References

This appendix contains the references. B Index

This appendix contains the index.