Complex Analysis. PH 503 Course TM. Charudatt Kadolkar Indian Institute of Technology, Guwahati

Complex Analysis PH 503 Course TM 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 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: Algebraic Properties In the rest of the chapter use. /. 0 /. 1 / 2 2 2 for complex numbers and3 4 5 for real numbers. introduce6 and7 8 9 : 6 ; notation. 1. Commutativity 2. Associativity 3. Distributive Law 7 < : 7 = 8 7 = : 7 < > 7 < 7 = 8 7 = 7 <? 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 4. Additive and Multiplicative Indentity R Q P Q X Y Z 5. Additive and Multiplicative Inverse C F C D E C G H B C C D E C C G C E J K L M L N O P Q S[ R T U \ R V W ] ^ _ ` ^ a ] ^ b_ ` ^ c d e gf h

ã ð å ó ó î 2 Chapter 1 Complex Numbers 6. Subtraction and Division i j k i l m i j n o k i l p q 7. Modulus or Absolute Value st s u v w x y z x i j i l m i j i lr j 8. Conjugates and properties { w } ~ z w } z { ƒ ƒ ˆ 9. Š Š Œ Ž 10. Triangle Inequality žÿ ž žÿ ž š œ Polar Coordinates and Euler Formula Ÿ 1. Polar Form: for, Ÿ ª «± ² where³ µ µ and ¹ º» ¼ ½ ¾. is called the argument ofà. Since Á Â Ä is also an argument of Å Æ the principle value of argument of Å is take such that Ë Ì ForÍ Î Ï theð Ñ Ò Ó Ç Ä È É Ê is unde ned. 2. Euler formula: Symbollically, Ô ÕÖ Ø Ù Ú Û Ü Ý Þ Û ßà Ü á 3. â ã â ä å ä æ ç èé ê ë ì í î ï ð ï ñ ò ñ ô õ ö ø ù ï û ü ý û þ û ÿ 4. de Moivre s Formula û ü

Roots of Complex Numbers Roots of Complex Numbers 3 Let then "! # # There are only " distinct roots which can be given by " ) If $ % & ' & ( ( ( & ' ( * is a principle value of+, -. then* / " is called the principle root. Example 1.1 The three possible roots of 0 142 53 6 7 8 9 : ; < =>? @ A B C D aree FG C B H I E FG C B H J FH G C D I E FG C B H J FK G C D. Regions in Complex Plane 1. L M N O P ofq R is de ned as a set of all pointss which satisfy TS U S R T V W 2. X Y Z [ \ ] ^ _ ` a ofb c is a nbd ofb c excluding pointb c. 3. Interior Point, Exterior Point, Boundary Point, Open set and closed set. 4. Domain, Region, Bounded sets, Limit Points.

É ¼ 2 Functions of Complex Variables Functions of a Complex Variable Ad e f g h ij k l de ned on a set m is a rule that uniquely associates to each pointn of m a a complex numbero p Setq is called ther s t u vw ofx andy is called the value ofx atz and is denoted byx { z } y ~ x { z } x { } ƒ ˆ Š Œ Ž Œ Œ Œ Ž š Œ Ž œ ž Ÿ Example 2.1 Writeª «± in² ³ µ form. ² «¾ ¹ ½º À» ¼¼½½Á ½ andâ Ä Å Æ Ç È ¾ Ê ½ À ¼ ½ Á ½ Ë Å Ì Ç È Ë Í É Î Ï Ð Ñ Ì andâ Ë Å Ì Ç È Ò Ë Í É Ð ÓÔ Ñ Ì Domain of Õ is Ö Ø Ù Ú. AÛ Ü Ý Þ ßà 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 û ü. One can choose the function to be single-valued by specifying õ ö ý õ þ ÿ where is the principal value. Elementary Functions

7 7 7 7 7 7 7 < < S 7 3 7 3 7 3 P P 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 function. Easy to see that $ % & ' ( ) * +, * -. +. Then $ / ' $ 0 1 ( ) * 2, * -. 2 3 a. $ / 4 $ / 5 ' $ / 4 6 / 5. b. $ / 6 7 8 % ' $ / # c. A line segment from 1 9 : ; 3 to 1 9 : < = 3 maps to a circle of radius $ 0 centered at origin. d. No Zeros. 3. Trigonometric Functions De ne ( ) * " ' @ A * -. " '. " ' $ % / $ > % / $ % /? $ > % / * -. " ( ) * " a. * -. 7 " ( ) * 7 " ' B b. < * -. " C ( ) * " ' * -. 1 " C " 3 * -. 1 " C? " c. < ( ) * " C ( ) * " ' ( ) * 1 " C " 3 ( ) * 1 " C? " d. < * -. " C * -. " '? ( ) * 1 " C " 3 ( ) * 1 " C? " e. * -. 1 " < = 3 ' * -. " and ( ) * 1 " < = 3 ' ( ) * " # f. * -. " ' ; iff " ' D = 1 D ' ; : E B : # # # 3 g. ( ) * " ' ; iff " ' 8 D = 1 D ' ; : E B : # # # 3 h. These functions are not bounded. i. A line segment from 1 ; : 2 3 to 1 < = : 2 3 maps to an ellipse with semimajor axis equal to ( ) * F G under H I J function. 4. Hyperbolic Functions De ne K L H F M N O H I J F M N O P Q O R P T S O R

h Mappings 7 a. U V W X Y Z [ \ ] Z U V W [ ^ _ ` a b c d e f g _ ` a e b. _ ` a b h e i a j k b h e g l c. a 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 d. a j k b e g p iff e g q o d c q g p r s l r t t t f e. _ ` a b e g p iff e g u v m q o w d c 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 function. b. Priniciple value of log function is given by Ž Ž Œ 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 Typically 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.

8 Chapter 2 Functions of Complex Variables 2 1.5 1 0.5-0.6-0.4-0.20 0 0.2 0.4 0.6 0.8 1 Mapping of Á Â Mapping Á Â 1. A straight line 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 Ö Ì 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 îí ï 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. ð ñ ò ð ñ ó ô A horizontal strip enclosed between and maps to the entire complex plane. 5. Sine Function õ ö ø ñ õ ö ù û õ ü ð ý þ 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.

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 Example 3.1! " # Show that $ % & ' ( ) * +, -. / 0-1 2 Example 3.2 +, -. / - 3 2 Show that 4 5 6 7 8 7 * +, -. / - 3 1 2 Example 3.3 +, -. / - 9 -: 2 Show that the limit of + does not exist as - ; <. Theorem 3.1 Let +, -. / =, >? @. A B C, >? @. and D 1 / = 1 A B C 1 2 7 4 8 5 6 7 * and only if E F G H I 4 5 6 8 E F * G H * I = / = 1 and E F G H I 4 5 6 8 E F * G H * I C / C 1 2 +, -. / D 1 if Example 3.4 +, -. / J 5 K -. Show that the 4 5 6 7 8 7 * +, -. / J 5 K - 1 Example 3.5 +, -. / L > A B @ 3. Show that the 4 5 6 7 8 3 M +, -. / N B. Theorem 3.2 If 7 4 8 5 6 7 * +, -. / D 1 and 7 4 8 5 6 7 * O, -. / P Q R 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 j k j l m n o p r q n o p a b c e c e c as 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:

~ ~ 12 Chapter 3 Analytic Functions 1. v w x y z y { } ~ 2. y v z w x 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 y v z w x 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 Ž Œ Ž Œ Ÿ Ÿ 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 ± real analysis ¹ º ¹ is not differentiable but ¹ º ¹» is.. In If a function is differentiable at ¼, then it is continuous at ¼.

Õ 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 differentiable at À be 1. Á Á Â 2. Á Á Â 3. Á Á Â 4. Á Á Â 5. Ù Ù Ú Û 6. Ù Ù Ú ½ Ä ¾ Å ½ ¾ Å ½ É ¾ Å Å Æ ½ Ç Å Æ ½ Ç Å Æ ½ Ç Å ¾ Å Ä ¾ Å ¾ Å È ½ Å À Å Ê ½ ½ Ð Ñ Ò Ó Ô Ò Õ Ö Ó Ñ Ó Ô Ò Ò Ñ Ó Ô Ò Ø Ü Ø Ô Ý Õ Þ Ô Ý ß à Ø Å ¾ Ç Å ¾ Ç Å À Å Å É Ë ¾ Å Ì Í if ¾ Å ÆÎ Ï À 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 Ö Ó Ô Ò Õ Ô ê Õ ã ê é å ê ë ì í ã å Ø Example 3.8 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 Show that the CR conditions are satis ed at û ü ý but the function þ is not differentiable at ý

14 Chapter 3 Analytic Functions If we write ÿ then we can write Cauchy-Riemann Conditions in polar coordinates: Analytic Functions De nition 3.4 in that set. A function is analytic in an open set if it has a derivative at each point Harmonic Functions 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. 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 1 2 3 4 5 6 7 0 8 2 3 4 5 6 9 : ; 2 = 6 9 > 2 3 4 5 6 7? @ 2 3 4 5 6 Theorem 3.5 If a function < the functionsb andc are harmonic in A. is analytic in a domain A then De nition 3.8 If two given functions B D E F G H and C D E F G H are harmonic in domain D and their rst order partial derivatives satisfy Cauchy-Riemann Conditions B I J C K B K J L C I M thenc is said to be N O P Q R S T U U R S V W X Y Z [ of\ ] Example 3.15 not vice versa. Let \ ^ _ ` a b c _ d e a d and f ^ _ ` a b c g _ a ] Show that f is hc of \ and Example 3.16 \ ^ _ ` a b c a h e i _ a. Find harmonic conjugate of\ ]

4 Integrals Contours Example 4.1 equations. Represent a line segment joining points j k l m n and o p q r r n by parametric 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 wherev 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 ž curve cuts itself and is closed. An arc is called simple if Ÿ An arc is closed if ˆ ˆ. Ž Ÿ ˆ Ž ˆ ƒ. Show that the An arc is differentiable if ˆ ˆ ˆ exists and ˆ and ˆ are continuous. A smooth arc is differentiable and ˆ is nonzero for all. De nition 4.2 ª «± ² ³ µ ² Length of a smooth arc is de ned as The length is invariant under parametrization change.

Ó Ó Ö Ö Ö 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 Ä ¾ À Å Æ Ç ¾ À and a function È ¾ ½ À É ¾ Ä Ê Ç À Å Æ Ë ¾ Ä Ê Ç À È is de ned on it. The integral of ¾ ½ À 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 Ô Õ Ï Ù Ú Û Ü Ý Ñ Ò Ø Þ ß Ô Õ Ï Ù Ý Þ Ü Ú Ñ Ò Ø 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 whereâ 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 ïð ñ ò ó ô õ ö ô ï ï ð ñ øò ó ô ó ù õ õ ô ó ù õ ø ö ù û 5. If øò 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 semicircular path in upper plane given by Example 4.6 Show that for and ÿ from to

B ' = C 2 ' 2 2 4 2 2 3 4 3 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 domain % 3 If $ is any simple closed contour in %, then & ' ( 4 +, 4. 5 3 ( 4 +. 4 6 " 7 8 9 ( 4 + " : ; < ( 4 + etc are entire functions so integral about any Example 4.7 2 loop is zero. Theorem 4.4 Let $ = and $ 6 be two simple closed positively oriented contours such that $ 6 lies entirely in the interior of $ = 3 If 2 is an analytic function in a domain % that contains $ = and $ 6 both and the region between them, then & ' > ( 4 +, 4. & '? ( 4 +, 4 ( 4 Example 4.8 2 +. @ A 4 ( 4. Find B +, 4 if $ is any contour containing origin. Choose a circular contour inside $ 3 Example 4.9 C D C E, 4. F G H 4 I if $ contains ' 6? K Example 4.10 Find B C J C6 where $ L M M. F 3 Extend the Cauchy Goursat theorem to multiply connected domains. Antiderivative 4 I 4 I 4 " then the function N ( 4 +. & ' ( 4 +, 4 Theorem 4.5 (Fundamental Theorem of Integration) connected domain % and is analytic in %. If and contour in% joining and Let 2 be de ned in a simply are points in % and $ is any

ba b Z Z c c c i c c 18 Chapter 4 Integrals is analytic in O andp Q R S T U V R S T W De nition 4.4 If V is analytic in O and S X and S Y are two points in O then the de nite integral is de ned as \ Z [ Z ] V R S T ^ S U P R S Y T _ P R S X T where P is an antiderivative of V. Ybc` a S Y ^ S U S d e f gy ` a g Example 4.11 Example 4.12 Example 4.13 U Y d h i X X d W X j k l S U l mn S oax U l m n i _ l m n p W Z [ Z ] qz U rk s S Y _ rk s S X W Cauchy Integral Formula Theorem 4.6 (Cauchy Integral Formula) Let V be analytic in domaino. Let t be a positively oriented simple closed contour in O. If S is in the interior of t then \ w V R S T U u vp V R S T ^ cs S _ S W Example 4.14 V R S T U Z [ X` x W b w Find V R S T ^ S if t y os _ i o U u W Example 4.15 V R S T U Y Z ` X W b w Find V R S T ^ S ift is square with vertices on R z u { z u T W Theorem 4.7 at that point. If V is analytic at a point, then all its derivatives exist and are analytic V } ~ R S T U u v i \ w V R S T ^ S R S _ S T } ` X

Ý ÖÌ 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 if and only if De nition 5.2 a series and is denoted by De nition 5.3 A series Ü sums converges to ç. Suppose that ª «µ ª ± ² ª and «µ ± ² Then, ¹ ª º» ¼ ½ ¾ ¼ ÀÁ ½ » ½ ¾ Ä Å Æ Ê ÇËÈ É Ì Í Ê Î Í If Ï Ð Ê Ñ is a sequence, the in nite sumð Ò Ó Ð Ô Ó Õ Õ Õ Ó Ð Ê Ó Õ Õ Õ is called. Ø Ù Ú Û is said to converge to a sum Û Ø Ù Ú Û Þ if a sequence of partial Þ ß à á â ã á ä ã å å å ã á æ

ù ù ö ý ý ý ý ý ö ý ö õ ö ý ö ý ö ý 20 Chapter 5 Series Taylor Series Theorem 5.2 if and only if Example 5.6 Suppose that è é ê ë é ì í î é and ï ê ð ñ ò ó ô Then, ø ù û ü ý ø ù þ û ÿ û if ø ù û Laurent Series Theorem 5.3 (Taylor Series) If is analytic in a circular disc of radius andcentered at then at each point inside the disc there is a series representation for given by û ø where û Example 5.7 û ù Example 5.8 ù û Example 5.9 û Example 5.10 û Theorem 5.4 (Laurent Series) such that ù for given by where If is analytic at all points in an annular region then at each point in there is a series representation û ø ø ù û # $ %! " ù û # $ %! " & ù

Laurent Series 21 and ' is any contour in (. 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. / 0 1. Example 5.12 2 32 4 0 5 / 6 7 2 8 9 2 > :?; < = where @A @ B C D Example 5.13 E F A G H K > I JIL K J> I M L D Find Laurent series for all @A @ N C O C N @A @ N P and @A @ B P D Example 5.14 Note that Q J H M RJ S T U E F A G V A D

6 Theory of Residues And Its Applications Singularities De nition 6.1 If a function W fails to be analytic at X Y but is analytic at some point in each neighbourhood of X Y, then X Y is az [ \ ] ^ _ ` a b c [ \ 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 jf k f g j i l l m for some then e is said to be an[z c _ ` d n o p qr 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 singularity 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.

Ç A Í Ç Í È Í 7 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  ½ such that is analytic for all in deleted nbd ½ Ä Å Ä Á. Then has a Laurent series representation The coef cient µ È ÆÉ È Ê µ Å È Ë È ÆÉ Ì µ Å È ¾ Ì Ï ÐÎ 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 Ö Ì Ö Ø. Then Ù Ó µ Ô Ï Ð Ñ Ì Ï Ð Ñ if Õ contains, otherwise½. Example 6.7 µ Ö Ú Ö Ì Ö Ø Û Ü Ý Show Þ ß à á â ã ä â å æ çé è if ê ë ìí î ï ì ð ñ ò Example 6.8 ó ô í õ ð í ö ø ù û ü. Show ý þ ó ô í õ ÿ í ð if ê ë ìí ì ð ñ ò Example 6.9 ó ô í õ ð ö ø ù û ü. Show ý þ ó ô í õ ÿ í ð if ê ë ìí ì ð ñ even though it has a singularity at í ð ò Theorem 6.1 ê, except for a nite number of points í í ò ò ò í inside, then Res If a function ó is analytic on and inside a positively oriented countour Example 6.10 Show that! " # $ % & ' Residues of Poles Theorem 6.2 If a function( has a pole of order) at " * then 9 : ; < Res( + " *, # -./ 0 1 2 4 5 6 3 9 = : ; < 4 4 = 6 = > :? 4 = 3 7 8 A Example 6.11 B C D E F H IG H J Simple pole ResB C D K E F L @ A 7 7

Quotients of Analytic Functions 25 Example 6.12 M N O P Q S TS UR S V WX. Simple pole at O Q Y Z Res M N Y P Q [ \ [ ]. Pole of order^ ato Q _. ResM N _ P Q ` [ \ [ ]. Example 6.13 M N O P Q d e fd a b gc d. h i j Resk l m n o p q r s t. Resk 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 functionk 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.

A References This appendix contains the references.

B Index This appendix contains the index.