COMPLEX ANALYSIS-I. DR. P.K. SRIVASTAVA Assistant Professor Department of Mathematics Galgotia s College of Engg. & Technology, Gr.

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1 COMPLEX ANALYSIS-I DR. P.K. SRIVASTAVA Assistant Professor Department of Mathematics Galgotia s College of Engg. & Technology, Gr. Noida An ISO 9001:2008 Certified Company Vayu Education of India 2/25, Ansari Road, Darya Ganj, New Delhi

2 COMLEX ANALYSIS I Copyright VAYU EDUCATION OF INDIA ISBN: First Edition: 2013 Price: 150/- All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Authors and Publisher. Printed & bounded in India Published by: (An ISO 9001:2008 Certified Company) VAYU EDUCATION OF INDIA 2/25, Ansari Road, Darya Ganj, New Delhi Ph.: , Fax: vei@veiindia.com

3 Preface

4 About The Book

5 Syllabus Math 1.3 : Complex Analysis I Block I : Complex Numbers. 1. Arithmetic operations. Conjugation, Absolute value, Inequalities. 2. Geometric addition and multiplication. the binomial equation, the spherical representation. 3. Lines and Circles. 4. Limits and Continuity. Block II : Analytic functions and power series. 1. Analytic functions, Cauchy Riemann equations. Harmonic functions, rational functions. 2. Elementary theory of power series, sequences, uniform convergence, Abel s limit theorem. 3. The exponential, logarithmic and trigonometric functions. 4. Toplology of the complex plane. Linear transformations, elementary conformal mappings. Block III : Comple Integration. 1. Line integrals, rectifiable arcs 2. Cauchy s theorem for a rectangle. Cauchy s theorem in a disc. 3. The index of a point with respect to a closed curve. Cauchy s integral formula, Liouville s theorem. The fundamental theorem of Algebra. 4. Removable singularities, Taylor s theorem, zeros and poles, The maximum principle.

6 Contents Preface...v BLOCK 1 1. Arithmetic operations Introduction Complex Numbers Equality of Complex Numbers Complex Conjugate Graphical Representation of a Complex Number Absolute of the Complex Number Conjugation Inequalities Properties of the Arguments Geometric Operations Geometric of complex numbers Subtraction of Complex Numbers Multiplication of Complex Numbers Division of Complex Numbers Some Properties of Complex Numbers Polar Form of Complex Numbers Product of Complex Numbers In Polar Form Quotient of Complex Numbers In Polar Form The Binomial Equation The Sperical Representation...20

7 3. Lines and Circles Complex Equation of A Straight Line and Circle Polynomials Behaviour of a Polynomial at Infinity Limit and Continuity Neighbourhood of Point Limit of a Function Some Theorems on Limits Continuous Function...28 BLOCK 2 1. Analytic Functions Analytic Function Cauchy Riemann Equation Cauchy s-riemann Equations In Polar Co Ordinates Harmonic Functions Milne - Thomson's Method Construction of Analytic Function Application of Analtic Functions To Flow Problems Rational Functions The Poles of The Derived Function of A Rational Function Poles and Zeros of A Rational Function at Infinity Normal Form of A Rational Function Power Series Power Series Absolute Convergence of Power Series Sanzn Tests for Convergence of Series Circle of Convergence and Radius of Convergence Sequences Limit Inferior and Limit Superior of Real Sequences Uniform Convergence Transcendental Functions Introduction Function of a Complex Variable Exponential Function of a Complex Variable...71

8 3.4 Properties of Exponential Functions Trigonometric Euler's Theorem Periodicity of Circular Functions Trigonometrical Formulae for Complex Variables Logarithms of Complex Quantity To Show That is Many Valued Function Logarithm of a Positive Real Number i.e. N Logarithm of a Negative Number i.e. [ N] Logarithm of a Purely Imaginary Number i.e. [ib] To Prove that log To Separate log (a + ib) into Real and Imaginary Parts Laws of LogarIthms for Complex Numbers The General Exponential Function Conformal Mappings Trnasformation or Mapping Conformal Mapping or Geometrical Representation of w = f(z) Coefficient of Magnification and Angle of Rotation Bilinear Transformation or Mobius Transformation or Fractional Trans Formation...97 BLOCK 3 1. Line integrals Simply and Multiply Connected Domain Partitions Rectifiable Are Riemanns Integration Green s Theorym in the Plate One Integrals Reduction of Complex Integrals to Real Integrals Properties of Complex Integrals Property An Estimation of Complex Integrat Cauchy s Theorems Cauchy s Theorem for a Rectangle...129

9 2.2 Cauchy s Theorem in Circular Disc State and Prove Cauchy s Integral Theorem Cauchy s integration Index of a Point with Respect to a Closed Curve: State and Prove Cauchy Integral Formula Liouville s Theorem Fundamental Theorem of Algebra Singularities Singular Points Isolated Singular Point Non-isolated Singular Point Types of Singularities Test for Detecting Singularities Series of Complex Terms Power Series, Circle of Convergence, Radius of Convergence Taylor's Theorem Maximum principle Index

10 BLOCK-1 Chapter 1: Arithmetic Operations Chapter 2 : Geometric OperationsfControl Systems Chapter 3 : Lines and circles Chapter 4 : Limit and Continuity

11 CHAPTER 1 ARITHMETIC OPERATIONS Introduction to Mechatronics 1.1 INTRODUCTION You may recall that the solution of the quadratic equation ax 2 + bx + c = 0, a 0 with real coefficient a, b, c is given by real number x 1 and x 2, where x 1 = a x a only if b 2 4ac 0. For b 2 4ac < 0, we do not have real solution of above quadratic equation. The mathematical to have solution for negative discriminant led us to extend the real number system to a new kind of numbes, namely, complex numbers that allow the square root of negative numbers. Let us consider solution of simple quadratic equation x = 0, its solution are x = +2 1 We assume that square root of 1 is denoted by symbol l called imaginary unit. Thus, for any two real numbers a and b, we can form a new number a + ib. This number a + ib is called a complex number COMPLEX NUMBERS Given a complex number a + ib, a is called its real part and b its imaginary part. For example 2 i, is a complex number in which 2 is a real part and j is the imaginary part. Remarks (i) A complex number is denoted by a single letter such as z, n etc. (ii) For any complex number z = a + ib, real and imaginary part of z are denoted as Rez = a. Imz = b (iii) If b = 0, then z = a + i0 = a is a purely real number and if a 0, then z = 0 + ib + ib is a purely imaginary number.

12 4 COMPLEX ANALYSIS I 1.3. EQUALITY OF COMPLEX NUMBERS Two complex numbers z 1 = a 1 + ib 1 and z 2 = a 2 + ib 2 are said to be equal if their real and imaginary parts are separately equal. In othre words z 1 = z 2 if and only if a 1 = a 2 and b 1 = b COMPLEX CONJUGATE If z = a + ib, the number a + ib is called the complex conjugate for simple conjugate of a = ib and is denoted by 5. For example 3 + i is the complex congugate of 3 + i GRAPHICAL REPRESENTATION OF A COMPLEX NUMBER A point in the XOY plane is uniquelly determined by its x and y co-ordinates. i.e., by an ordered pair (x, y) of real numbers. Complex number may be assigned to points in the plane in the same way as ordered pairs of real numbers. Fig. 1.1 Fig. 1.2

13 ARITHMETIC OPERATIONS 5 Thus ever complex number x + iy can be represented geometrically as a unique point P(x, y) in the XOY plane with x co-ordinate representing its real part and y co-ordinate representing its imaginary part. The point (x, 0) on the x-axis represent complex number x + i 0. i.e., the real number x and every real nmber is represented as a point on this axis. As such x-axis is called the real axis. Similarly, the point (0, y) on the y-axis represents complex numbers 0 + iy, i.e. the imaginary number iv. Therefore, y-axis is called the imaginary axis ABSOLUTE OF THE COMPLEX NUMBER The distance from the origin to the point P(x, y) is defined as the modulus (or absolute value) of the complex number z and x + iy and is denoted by z, i.e., Fig. 1.3 Note If z = x + iy, then Z = 0 = Argument of z (bx + iy) = tan y x (i) (ii) (iii)

14 6 COMPLEX ANALYSIS I Example 1.1 Represent the complex number 2 + 3i by a point in the complex plane. Solution: The complex number 2 + 3i is represented by a point with x-co-ordinate = Re (2 + 3i) = 2 and y-co-ordinate = lm (2 + 3i) = 3. The point P (2, 3) is located by 2 units on the positive x-axis of real numbrers and 3 units onthe positive y-axis of imaginary numbers CONJUGATION Fig 1.4 If z = x + iy, then the complex number x iy is called the conjugate of the complex number z and is writtenn as z. It is easily seen that number conjugate to z 1 + z 1 and z 1 z 2 are and zz 1 2respectively. Also we have z 2 = z, 2R (z) = z + z and 2 ii (z) = z z. It is clear that z = z Fig. 1.5

15 ARITHMETIC OPERATIONS 7 Geometrically, the conjugate of z is the reflection for (image) of z in the real axis INEQUALITIES Theorem 1.1 The modulus of the sum of two complex numbers can over exceed the sum of their moduli. Proof. We shall prove that z 1 + z 2 z 1 + z 2 We have z 1 + z 2 2 =(z 1 + z 2 ) ( z 1 + z 2 ) = z 1 z 1 + z 1 z 2 + z 2 z 1 + z 2 z 2 = z 1 z 1 + z 2 z 2 + (z 1 z 2 + z 1 z 2 )...(i) Now z 1 z 2 = z 1 2, z 2 z 2 = z 2 2 and z 1 z 2 + z 1 z 2 =(x 1 + iy 1 ) (x 2 iy 2 ) + (x 1 iy 1 )(x 2 + iy 2 ) =2 (x 1 x 2 + y 1 y 2 ) = 2R (z 1 z 2 ) 2 z 1 z 2, Since the real part of a complex number can never exceed its modulus. Then (i) gives z 1 + z 2 2 = z z R (z 1 z 2 ) since zz 1 2 = z 1 z 2, z 2 + z z 1 z 2, = Hence. Remark. Since = 1 2 we have by addition, 2 z z = Alternative Method Writing z 1 = r 1 (cos 1 + i sin 1 ) and z 2 = r 2 (cos 2 + i sin 2 ), we get z 1 + z 2 = (r 1 cos 1 + r 2 cos 2 ) + i(r 1 sin 1 + r 2 sin 2 ); so that Ö = [r r r 1 r 2 ] [ cos ( 1 2 ) 1]

16 Complex Analysis By Dr. P.K. Srivastava Publisher : Vayu Education ISBN : Author : Dr. P.K. Srivastava Type the URL : 2 Get this ebook