3_Complex_Numbers

3_Complex_Numbers - Chapter 3 Review of Complex Variables...

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35 Chapter 3 Review of Complex Variables Chapter Outline 3.1 COMPLEX NUMBERS. .......................................................................... 36 3.1.1 Introduction . .................................................................................................................................... 3.1.2 Definitions. ...................................................................................................................................... 3.1.3 Graphical Representation. ........................................................................................................ 37 3.1.4 Sets in the Complex Plane. ..................................................................................................... 41 3.2 COMPLEX FUNCTIONS. ........................................................................ 42 3.2.1 Definitions. 3.2.2 Rational Functions . ..................................................................................................................... 43 3.2.3 Second-Order Polynomials. 45 3.3 HOMEWORK FOR CHAPTER 3. ............................................................ 49 This chapter contains a review of complex numbers and complex rational functions. This material is required throughout the text. For the most part, this material is found in all introductory treatments of complex variables. The notation is standard. The algebraic and geometric representation of complex numbers and complex arithmetic are routinely used throughout the text. The exponential representation is fundamental to the algebra of complex numbers. Complex exponentials also play a central role in Fourier series. This material is discussed in Section 3.1. Rational functions of a complex variable play a central role for Laplace and z - transforms. Insofar as these two subjects play a central role in continuous and discrete systems, this material is absolutely essential to obtain a firm grasp of the concepts developed in the remainder of the text. Because of the commonality of the concepts of Laplace and z -transforms, basic terminology of rational functions are presented in Section 3.2. In particular, the indexing convention of the coefficients of the rational function is followed throughout the discussion of continuous-time systems. This section also contains a discussion of second-order polynomials that is particular to certain engineering subjects. Summary of Sections Section 3.1: We present a review of complex numbers. Section 3.2: We discuss rational functions. Coverage of the Text This chapter is self-contained.
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36 Chapter 3 Review of Complex Variables 3.1 COMPLEX NUMBERS 3.1.1 Introduction We begin with the definition of a complex number along with the simple arithmetic properties of these numbers. We present two representations of a complex number which are used interchangeably. These representations of a complex number have a graphical representation as well, which is developed below. 3.1.2 Definitions Definition 3.1.1 : Let j () =- 1 be the root of the equation s 2 10 += . A complex nu mber s is defined as sj =+ σϖσ ϖ ,, RR . ▲▲ Notation : The set of complex numbers is denoted by C . The components parts of a complex number are as follows. Definition 3.1.2 : The rectangular representation of the complex number s is σϖ . The real part of s is σ, write Re( ) . s The imaginary part of s is ϖ, write Im( ) . s The algebraic manipulations of complex numbers are defined as follows. Definition 3.1.3 : We define addition (and subtraction) of two complex numbers as ss j j j 1 211 2 2 1 2 1 2 += + + + (29 ++ . σ σ ϖ ϖ We define the multiplication of two complex numbers as () ( ) . j j j 1 2 1 2 12 + σ σ ϖ ϖ ϖ σ σ ϖ We define the division of two complex numbers as s s j j j 1 2 11 22 2 2 2 2 2 2 2 2 = + + = + + + - + σσ ϖϖ ϖσ .
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Section 3.1 Complex Numbers 37 Example 3.1.4 : Consider the complex number sj 0 14 =+ . The real part and imaginary part of s 0 , respectively, are Re( ) Im( ) .
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This note was uploaded on 09/10/2009 for the course ECE 60367 taught by Professor Meehan during the Spring '09 term at Virginia Tech.

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3_Complex_Numbers - Chapter 3 Review of Complex Variables...

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