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M.Sc. (Mathematics), SEM- I Paper - III COMPLEX ANALYSIS

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M.Sc. (Mathematics), SEM- I Paper - III COMPLEX ANALYSIS M.Sc. (Mathematics), SEM- I Paper - III COMPLEX ANALYSIS PSMT103 CONTENT Unit No. Title 1 INTRODUCTION TO COMPLEX NUMBER SYSTEM 2 SEQUENCES OF COMPLEX NUMBERS 3 SERIES OF COMPLEX NUMBERS 4 DIFFERENTIABILITY 5 COMPLEX LOGARITHM 6 ANALYTIC FUNCTIONS 7 COMPLEX INTEGRATION 8 CAUCHY THEOREM 9 THEOREMS IN COMPLEX ANALYSIS 10 MAXIMUM AND MINIMUM MODULUSPRINCIPLE SINGULARITIES 11 RESIDUE CALCULUSAND MEROMORPHIC FUNCTIONS 12 MOBIUS TRANSFORMATION *** SYLLABUS Unit I. Holomorphic functions Note: A complex differentiable function defined on an open subset of is called a holomorphic function. Review: Complex numbers, Geometry of the complex plane, Weierstrass’s M-test and its aplication to uniform convergence, Ratio and root test for convergence of series of complex numbers. Stereographic projection, Sequence and series of complex numbers, Sequence and series of functions in , Complex differential functions, Chain rule for holomorphic function. Power series of complex numbers, Radius of convergence of power series, Cauchy-Hadamard formula for radius of convergence of power series. Abel's theorem: let be a power series of radius of convergence R > 0: Then the function f(z) defined by is holomorphic on the open ball and for all : Trigonometric functions, Applications of Abel's theorem to trigonometric functions. Applications of the chain rule to define the logarithm as the inverse of exponential, branches of logarithm, principle branch of the logarithm and its derivative on Unit II. Contour integration, Cauchy-Goursat theorem Contour integration, Cauchy-Goursat Theorem for a rectangular region or a triangular region. Cauchy’s theorem(general domain), Cauchy integral formula, Cauchy’s estimates, The index(winding number) of a closed curve, Primitives. Existence of primitives, Morera’s theorem. Power series representation of holomorphic function (Taylor’s theorem). Unit III. Properties of Holomorphic functions Entire functions, Liouville’s theorem. Fundamental theorem of algebra. Zeros of holomorphic functions, Identity theorem. Counting zeros; Open Mapping Theorem, Maximum modulus theorem, Schwarz’s lemma. Automorphisms of unit disc. Isolated singularities: removable singularities and Removable singularity theorem, poles and essential singularities. Laurent Series development. Casorati- Weierstrass’s theorem. Unit IV. Residue calculus and Mobius transformation Residue Theorem and evaluation of standard types of integrals by the residue calculus method. Argument principle. Rouch´e’s theorem. Conformal mapping, Mobius Transformation. 1 1 INTRODUCTION TO COMPLEX NUMBER SYSTEM Unit Structure : 1.0. Objectives 1.1. Introduction 1.2. The Field of Complex Numbers 1.3. Extended Complex Plane, The Point at Infinity, Stereographic Projection 1.4. Summary 1.5. Unit End Exercises 1.0. OBJECTIVES: After going through this unit you shall come to know about The field of complex numbers denoted by ℂ. Representations of complex numbers in polar forms . 2 The Euclidean two dimensional plane along with the point at infinity forms the extended complex plane. The extended complex plane is in one to one 3 correspondence with the unit sphere in and such a correspondence is known to be the stereographic projection. 1.1. INTRODUCTION : Numbers of the form z a bi , where a and b are real numbers and i 1are called as Complex Numbers. The identities involving complex numbers lead to solutions to many problems in the theory of real valued functions .The wider acceptance of complex numbers is because of the geometric representation of complex numbers , which was fully developed and studied by Gauss. The first complete and formal definition of complex umbers was given by William Hamilton. We shall begin with this definition and then consider the geometry of complex numbers. 2 1.2. THE FIELD OF COMPLEX NUMBERS : A complex number z is an ordered pair x, y of real numbers. i.e. z x,, y x y . Complex number system, denoted by is the set of all ordered pairs of real numbers (i.e. ) with the two operations of addition and multiplication ( or ) which satisfy : (i) x1 , y 1 x 2 , y 2 x 1 x 2 y 1 y 2 x1,,, y 1 x 2 y 2 (ii) xy11 , . xy 22 , xx 12121221 yyxy , xy The word ordered pair means x1, y 1 and y1, x 1 are distinct unless x1 y 1. Let z x,, y x y .‘x’ is called Real part of a complex number z and it is denoted by x Re z , (Real part of z) and ‘y’ is called Imaginary part of z and it is denoted by y Im z . Two complex numbers z1 x 1, y 1 and z2 x 2, y 2 are said to be equal iff x1 x 2 and y1 y 2 i.e. real part and imaginary part both are equal. About Symbol ‘i’: The complex number 0, 1 is denoted by ‘i’ and is called the imaginary number. i2 i. i 0,1 . 0,1 0 1, 0 0 by property (ii) abov 1, 0 i2 Similarly, i3 i 2. i 1,0.0,1 00, 10 0,1 i3 i i4 i 3. i 0,1.0,1 01,00 1,0 3 i4 1 Using this symbol i, we can write a complex number x, y as x iy (Since x iy x,0 0,1 y ,0 x ,0 0, y x , y The complex number z x, y can be written as z x iy Note: (The set of all complex numbers) forms a field. 3 Propeties of complex numbers Let z1 x 1, y 1 , z2 x 2, y 2 and z3 x 3, y 3 . 1) Closure Law : z1 z 2 and z1. z 2 2) Commutative Law of addition : z1 z 2 z 2 z 1 zz1211 xy,,,, xy 22 xxyy 1212 xxyy 2121 x2,, y 2 x 1 y 1 z 2 z 1 3) Associative Law of addition : z1 z 2 z 3 z 1 z 2 z 3 z1 z 2 z 3 x 1,,, y 1 x 2 y 2 x 3 y 3 xyxxyy11,,, 2323 xxxyyy 123123 xxyy1212 ,,,, xy 33 xy 11 xy 22 xy 33 z1 z 2 z 3 4) Existence of additive Identity : The Complex Number 0 0, 0 i.e. z 0 0 i is called the identity with respect to addition. 5) Existence of additive Inverse : For each complex number z1 , a unique complex number z s.t. z1 z z z 1 0 i.e. z z1. The complex number z is called the additive inverse of z1 and it is denoted by z z1. 6) Commutative law of Multiplication : z1.. z 2 z 2 z 1 z.,., z x y x y x x y y, x y x y (1) 1 2 1 1 2 2 1 2 1 2 1 2 2 1 ….. and zz21.,.,.,.. xy 22 xy 11 xxyyxyxy 21212112 x1..,.. x 2 y 2 y 1 x 1 y 2 x 2 y 1 z1. z 2 from (1) 7) Associative Law of Multiplication : z1.... z 2 z 3 z 1 z 2 z 3 z1..,,.. z 2 z 3 x 1 y 1 x 2 y 2 x 3 y 3 x1,..,.. y 1 x 2 x 3 y 2 y 3 x 2 y 3 x 3 y 2 xxxyy12323 ...,.... yxxyy 12323 xxxxy 12332 yxxyy 12323 xxxxyyxxyyyyxxxxxy123123131123123132......, x2. x 3 . y 1 y 1 y 2 y 3 (*) 4 zzz123..,.,,,, xyxy 11 22 xy 33 xxyyxyxyxy 1212122133 xxx123 xyy 312 xxy 123 yyy 123 , xxy 132 xyy 213 xxx 123 yyy 123 z1 z 2. z 3 from (*) 8) Existence of Multiplicative Identity : z1.1 1. z 1 z 1 The complex number 1 1, 0 (i.e. z1 0 i ) is called the identity with respect to multiplication. 9) Existence of Multiplicative Inverse : For each complex number z1 0, there exists a unique complex number z in s.t. 1 z1. z z . z 1 1 i.e. z is called the multiplicative inverse of z1 1 1 complex number z1 and it is denoted by z or z . z1 Let z x, y and z1 x 1, y 1 z1. z 2 1 x, y x1 , y 1 1, 0 xx1 yy 1, xy 1 x 1 y 1, 0 x. x1 y . y 1 1 ……..(i) and x. y1 x 1 . y 0 …......(ii) Equation (ii) x1 - Equation (i) y1, we get 2 xx1 y 1 x 1 y0 2 xx1 y 1 yy 1 y 1 2 2 y x1 y 1 y 1 y y 1 (iii) 2 2 x1 y 1 Substitute equation (iii) in equation (ii) i.e. x. y1 y . x 1 0 y x y 1 x.. y y x 1 x x 1 1 1 1 2 2 1 2 2 x1 y 1 x1 y 1 y1 x x 1 2 2 x1 y 1 x y z 1, 1 2 2 2 2 x1 y 1 x 1 y 2 z is the multiplicative inverse of complex number z1 x 1, y 1 . 5 10) Distributive Law : z1 z 2 z 3 z 1.. z 2 z 1 z 3 Subtraction: The difference of two complex Numbers z1 x 1, y 1 and z2 x 2, y 2 is defined as : z1 z 2 x 1,, y 1 x 2 y 2 x1 x 2, y 1 y 2 z1 1 Division: It is defined by the equality z1. z 2 z2 0 z2 x y x.. x y y x y x y x,,, y 2 2 1 2 1 2 1 2 2 1 1 1 2 2 2 2 2 2 2 2 x2 y 2 x 2 y 2 x 2 y 2 x 2 y 2 Geometrical Representation of a Complex Number : Consider a complex number z x iy .
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