COMPLEX ANALYSIS.pdf - LESSON 1 INTRODUCTION AND REVISION...

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1 LESSON 1 INTRODUCTION AND REVISION OF PRIMARY RESULTS IN COMPLEX ANALYSIS 1.1 Introduction This book, Complex Analysis II is a second course in Complex Analysis. The reader might have some preliminary knowledge in Complex Analysis, however the important results required to study the Complex Analysis is given in this chapter. The reader is advi sed to study the preliminary results in details in the author’s Book on Complex Analysis. 1.2 Objectives of the lesson By the end of this lesson you will be able to Define a Complex Number Do arithmetic operations of complex numbers Define the conjugate of a complex number Represent complex numbers on a graph called Argand Diagram or Complex plane. Know polar form (or r, form) of a complex number. State De Moivre’s Theorem and apply the same for finding roots of real and complex numbers. Define single valued and many valued functions of complex variables. Define Limit of a function Define continuity of a function Define Analytic function at a point and in a region Define an entire function Explain Cauchy-Reimann Equations. Define simply and multiply connected Region Know Cauchy’s fundamental theorem 1.3 Definition of Complex Numbers A Complex number can be defined as a number of the form z = x + iy where x and y are real numbers and i is such that i 2 = -1 or i = 1 1.4 Arithmetic operations of Complex Numbers In performing operations with complex numbers we can proceed as in the Algebra of real numbers keeping in mind 1 i , i 2 = -1, i 3 = -i, i 4 =1 and i 5 =i and so on. Thus (3 + 4i) + (2 5i) = 5 i (3 + 4i) - (2 5i) = 1 + 9i (3 + 4i) (2 5i) = 6 15i + 8i 20i 2 = 26 7i
2 1.5 Conjugate of a complex number If z = x + iy is any complex number, we define the conjugate of the complex number z as iy x z . The reflection of z in x axis or real axis is defined as the conjugate of z and it

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