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CU-Boulder PHYS 3210 - Complex Numbers

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3Complex numbers andhyperbolic functionsThis chapter is concerned with the representation and manipulation of complexnumbers. Complex numbers pervade this book, underscoring their wide appli-cation in the mathematics of the physical sciences. The application of complexnumbers to the description of physical systems is left until later chapters andonly the basic tools are presented here.3.1 The need for complex numbersAlthough complex numbers occur in many branches of mathematics, they arisemost directly out of solving polynomial equations. We examine a specific quadraticequation as an example.Consider the quadratic equationz"-42*5:0.Equation (3.1) has two solutions, zy a.nd 22, such that(z - zr)Q - z) :0.Using the lamiliar formula for the roots of a quadraticsolutions z1 and 22, written in brief as 2r.2, are(3.1 )(3.2)equation, (1.4), the4+ J;$ -411" '(3.3)Both solutions contain the square root of a negative number. However, it is nottrue to say that there are no solutions to the quadratic equation. Thefundamentaltheorem of algebra states that a quadratic equation will always have two solutionsand these are in fact given by (3.3). The second term on the RHS of (3.3) iscalled an imaginary term since it contains the square root of a negative number;83COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONSFigure 3.1 The function f (r) -- "' - 4z * 5.the first term is called a real term. The full solution is the sum of a real termand an imaginary term and is called a complex number. A plot of the functionIQ): z2 -42 * 5 is shown in figure 3.1. It will be seen that the plot does notintersect the z-axis, corresponding to the fact that the equation /(z) :0 has nopurely real solutions.The choice of the symbol z for the quadratic variable was not arbitrary; theconventional representation of a complex number is z, where z is the sum of areal part x and i times an imaginary part y, i.e.z:x*iy,where i is used to denote the square root of - 1 . The real part x and the imaginarypart y arc usually denoted by Rez andlmz respectively. We note at this pointthat some physical scientists, engineers in particular, use j instead of i. However,for consistency, we will use i throughout this book.In our particular example, lZ :2J-t - 2i, and hence the two solutions of(3.1 ) arezrz:2tzj:zti.Thus, here x:2 and,y : *1.For compactness a complex number is sometimes written in the formz : (x,y),where the components of z may be thought ofas coordinates in an xy-plot. Sucha plot is called an Argand diagram and is a common representation of complexnumbers; an example is shown in figure 3.2.3.2 MANIPULATION OF COMPLEX NUMBERSFigure 3.2 The Argand diagram.Our particular example of a quadratic equation may be generalised readily topolynomials whose highest power (degree) is greater than2, e.g. cubic equations(degree 3), quartic equations (degree 4) and so on. For a general polynomial /(z),of degree n, the fundamental theorem of algebra states that the equation f(z) : Owill have exactly n solutions. We will examine cases of higher-degree equationsin subsection 3.4.3.The remainder of this chapter deals with: the algebra and manipulation ofcomplex numbers; their polar representation, which has advantages in manycircumstances ; complex exponentials and logarithms; the use of complex numbersin finding the roots of polynomial equations; and hyperbolic functions.3.2 Manipulation of complex numbersThis section considers basic complex number manipulation. Some analogy maybe drawn with vector manipulation (see chapter 7) but this section stands aloneas an introduction.3.2,1 Addition and subtractionThe addition of two complex numbers, 21 and 22, in general gives anothercomplex number. The real components and the imaginary components are addedseparately and in a like manner to the familiar addition of real numbers:4 + 22 :(xr * iyr) * (xz * iy) : (x1+x2) + i(y I yz),85lrn zCOMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS4+22Figure 3.3 The addition of two complex numbers.or in component notationz1 | z2 - (xr"yr) * 6z,y) : (xr * xz,yr * yz).The Argand representation of the addition of two complex numbers is shown infigure 3.3.By straightforward application of the commutativity and associativity of thereal and imaginary parts separately, we can show that the addition of complexnumbers is itself commutative and associative, i.e.z1*22:22+zt,zr*kz*4):Qrlzz)lzt.Thus it is immaterial in what order complex numbers are added.r sum the compl* nambas 1 + 2,, 3 - 4i, -2 + ,.Summing the real terms we obtainl+3-2:2.and summing the imaginary terms we obtain2i-4i+i:-i.(1 + 2D + (3 -4') + (-2 + i) : 2 - i. <The subtraction of complex numbers is very similar to their addition. As in thecase of real numbers, if two identical complex numbers are subtracted then theresult is zero.Hence863.2 MANIPULATION OF COMPLEX NUMBERSFigure 3.4 The modulus and argument of a complex number3.2.2 Modulus and argumentThe modulus of the complex number z is denoted by lzl and is defined asltl:\,Frt.(3.4)Hence the modulus of the complex number is the distance of the correspondingpoint irom the origin in the Argand diagram, as may be seen in figure 3.4.The argument of the complex number z is denoted by arg z and is defined asa,sz-*1l-'(i)(3.5)It can be seen that argz is the angle that the line joining the origin to z onthe Argand diagram makes with the positive x-axis. The anticlockwise directtonis taken to be positive by convention. The angle arg z is shown in figure 3.4.Account must be taken of the signs of x and y individually in determining inwhich quadrant arg z lies. Thus, for example, if x and y are both negative thenarg z lies in the range -n < arg z < -n/2 rather than in the first quadrant(0 < arg z <n/2), though both cases give the same value for the ratio of y to x.>Fuil tlw mdubs aulof the canplex rupiber z :2 -ii.Using (3.4), the modulus is given byEj: JY * 1-3Y : '81.Using (3.5), the argument is given byary z : tan-t (-)) .The two angles whose tangents equal -1.5 are -0.9828 rad and 2.1588 rad. Since x : 2 andy : -3, z clearly lies in the fourth quadrant; therefore atg z : -0.9828 is the appropriateanswer. <COMPLEX NUMBERS AND HYPERBOLIC FUNCTIONS3.2.3 MuhiplicationComplex numbers may be multiplied together and in general give a complexnumber as the result. The product of two complex numbers z1 &nd z2 is foundby multiplying them out in full and remembering that i2 : -1, t-e.z1z2 -- (x1 + iyt)(x2 + iy2): xfiz * ixflz + iyrxz + i2yryz: (4xz -


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