Calc3 Ch1 Complex Nos.pdf - Dr Janet Semester 2 2020/21...

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CALCULUS III Chapter 1: COMPLEX NUMBERS & FUNCTIONS Dr Janet Semester 2, 2020/21
1.1 Introduction 1.2 The Argand diagram 1.3 The arithmetic of complex numbers 1.4 Polar form of a complex number 1.5 Euler’s formula 1.6 Powers and roots of complex numbers 1.7 Circular and hyperbolic functions 1.8 Logarithm of a complex number 1.9 Complex functions and mappings 1.10 Complex differentiation. Cauchy-Riemann equations 1.11 Conjugate and harmonic functions 1.12 Power series. Taylor series 1.13 Laurent series 2
1.1 Introduction 3 We are familiar with “ordinary” numbers. E.g. 2 0. x Mathematicians call these real numbers. They can be plotted on a ‘real axis’ or ‘number line’. We know - how to add, subtract, multiply, divide them. - they can be positive, negative or zero. - for any real number x , In this chapter we consider a new set of numbers whose square is not necessarily positive! These are called complex numbers . 1, 2, 3/10, 2.634, , , 2, . e etc
4 Fundamental to complex numbers is the symbol j which has the property j is called the imaginary number *. Don’t worry about the fact that j doesn’t exist! We just define j then use it (in a similar way to real numbers) … and it is found to be extremely useful in a wide range of practical applications! e.g. a.c. circuits, fluid flow, heat transfer, signal processing. 2 1. j   *Note : Sometimes i is used instead of j . The Imaginary Number 1 j We assume we can write Hence By the usual rules of algebra, 4 3 j 4 j 7
5 Definitions A complex number is an ordered pair of real numbers, usually denoted by z (or w , etc.). For example, if x and y are real numbers then we can have a complex number where z x jy Examples: 2 1. j   For a complex number x is the real part of z , written y is the imaginary part of z , written E.g. for , z x jy 1 1 1 3 2 , Re 3, Im 2 z j z z Re x z Im y z If x = 0 then z = yj is said to be pure imaginary .
1.2 The Argand Diagram 6 A complex number involves a pair of real numbers x,y , so is naturally represented by a point in a plane. We take the usual Oxy plane but call it the complex plane . Ox is called the real axis , Oy the imaginary axis . A plot of the complex plane is called an Argand diagram . z x jy E.g. z = 2 + 3 j is represented by the point ( 2, 3 ). w = - 1 + j by (-1, 1 ).
1.3 Complex Arithmetic 7 Equality z 1 = z 2 if and only x 1 = x 2 and y 1 = y 2 . I.e. Two complex numbers are equal if and only if their real parts are the same and their imaginary parts are the same. 1 1 1 z x jy 2 2 2 z x jy Consider two complex numbers: Addition & Subtraction 1 2 1 2 1 2 ) ) ( ( z z x x j y y 1 2 1 2 1 2 ) ) ( ( z z x x j y y Note: this is just like for vectors!
Example 2 Given and , 8 a) Find , b) Find , c) Find 2 w , d) Plot z, w, 2 w, z + w and z w on an Argand diagram z w 3 2 z j 2 w j    z w
9 Multiplication Multiplication is done in the obvious way: multiply all the terms together and replace j 2 with - 1 wherever it occurs .

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