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A Basic Course in Complex Variables
By David C. Kay iUniverse
Copyright © 2014 David C. Kay
All rights reserved.
ISBN: 978-1-4917-4265-5
CHAPTER 1
COMPLEX NUMBERS AND COMMON NOTIONS
WORKING WITH IMAGINARY NUMBERS
You no doubt once solved quadratic equations having solutions that were imaginary, or complex. For example, consider x2 – 4x + 13 = 0. If you use either the method of completing-the-square or the quadratic formula, you come up with an answer of the form x = 2 [+ or -] 3i, where [square root of -1] = 1 . When you were first introduced to this, you were probably told (perhaps mysteriously to you) that i is an imaginary number whose square equals –1, and that it should be treated just like an ordinary "algebraic quantity". For example,
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and so forth. That is, all the ordinary rules of algebra are valid. (One might ask, "how do we know that for sure?")
Since we are going to be calculating with these kinds of numbers exclusively, it is good to review how they are supposed to work in elementary algebra. The steps in solving the above equation by completingthe-square are, as follows:
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To check this result, one substitutes 2 [+ or -] 3i for x, making use of the fundamental property i2 = –1. Thus (using just the one root x = 2 + 3i),
In general, and for future reference, we can list some basic identities involving this imaginary number i, all obtained by "ordinary algebra."
ALGEBRA AND IMAGINARY NUMBERS
Were you ever bothered by being told that the "number" i = [square root of -1] was to be treated as an ordinary algebraic quantity (like x), and to use the ordinary rules of algebra to work with it? For example, to find the product (2 + 3i) x (5 – i) you were instructed to just use the distributive law as in ordinary algebra. Never mind that [square root of -1] exists only in your imagination; certainly it does not exist in the "real world" we live in, and you will certainly not find it on an ordinary pocket calculator. Even assuming that i can be given some type of meaning, how can we be certain that the laws of algebra will not lead to a contradiction of some sort? Any such contradiction would be quite devastating.
Incidentally, the history of i is an interesting story of an evolving mathematical concept that engaged the minds of many great mathematicians of the past. They wrestled with these same questions. Leonhard Euler (1707–1783) was concerned about its true meaning in his book Algebra (1770), when he wrote: "Such numbers, which by their nature are impossible, are called imaginary or fanciful numbers [our italics] because they exist only in the imagination." It would take almost three centuries after they were first introduced (and used with great suspicion) before an adequate theory would be created that would embrace them in a wider "field of arithmetic" (Burton, 1991). The next section is devoted to such a field.
If you are willing to accept it on faith that no inconsistencies can arise in this very shallow treatment of complex numbers from high school, then you can skip the next section without missing a great deal. Just one thing though; consider the following example, which shows how the ordinary rules of algebra can run amuck. You might change your mind about skipping the next section. In this example, we assume that i exists, and that it equals [square root of -1].
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But this contradicts i2 = –1! Was any law of algebra violated? It seems not. So what went wrong?
A FOUNDATION FOR THE COMPLEX NUMBERS
In order to better understand the discussion which follows, it is helpful to take a broader view of what constitutes a "number." This is no different from the task you once had when negative numbers were introduced in elementary algebra; at that point, you had worked exclusively with positive numbers and zero, and then suddenly you had to learn how to work with a new kind of number. But any system of objects that obeys certain operational working rules can be regarded as a system of "numbers." The operations of addition (+) and multiplication (x) are defined in some unambiguous manner which must obey certain standard rules. These rules are familiar from elementary algebra—we assume you know what they are formally. They include the commutative laws of + and x (such as a x b = b x a), the associative laws, and the distributive law of multiplication, among others. Such laws are the attributes required in mathematics in order to define what is generally known as a field. One can regard the members of any such field as numbers because they behave the same way as the real numbers do.
Thus, the real number system with all its familiar working rules is a field, denoted by R. We aim to use this system to create a larger field which includes the number i, and more generally, the complex numbers a + bi. There are several ways to do this; we have chosen a system you should be familiar with, namely, 2x2 matrices of real numbers. In case this is new to you, let's describe what these objects are and how they work. This will at the same time provide review for others.
A real 2x2 matrix is merely a square array of numbers arranged in two rows and two columns. For example, the "numbers" in this system take on the appearance:
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where a, b, c, d, x, y, z, and w are real numbers. Next, as in a typical course in linear algebra, the sum and product of these matrices are defined:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The product rule is more complicated than we might expect, but there are good reasons for it, having to do with topics in linear algebra (which will be omitted). The important thing is that since we can add and multiply such matrices, they begin to act like "numbers". And familiar rules emerge. For example, the associative laws of addition and multiplication are valid. The distributive law of multiplication is also valid: for any three matrices A, B, and C, A(B + C) = AB + AC. We are not going to take the trouble to prove all these laws since this is normally covered in linear algebra courses (the problem section at the end of this chapter will consider a few of these proofs). But it is important to realize that for matrices in general, the ordinary rules of arithmetic apply. One exception: the commutative law of multiplication is not always valid. Examples abound, like the following:
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That is, AB is not always equal to BA, a significant departure from ordinary numbers. This property is required however, in order to obtain a field. What can be done about this? One way to solve this problem is to restrict the set of matrices we are willing to work with. For example, consider only matrices like
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If we multiply two such matrices in any order, we obtain
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By interchanging a and b,
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So the commutative law of multiplication is valid. What about the numbers "zero" and "one"? That's easy. Just define
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Now all the operational rules that are valid for real numbers are valid for this class of matrices. For example, A + O = A, OA = O and IA = A for any matrix A. But we have not gotten very far, because if the ordinary rule for scalar multiplication be adopted as it is in linear algebra, then all our matrices are of the form aI, for real a, and this just becomes a fancy way to display the real numbers, where the matrix aI represents the real number a. Thus instead of obtaining X2 = –I for some matrix X, we find that the square of a matrix in this system is just
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which represents the non-negative real number a2. Thus, squaring a matrix of this form does not produce a negative number (or one that we would regard as a negative number).
It takes much experimentation to come up with a different system that includes a matrix which we can regard as i. It was discovered a long time ago that the desired system we are seeking consists of all matrices of the form
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where a and b are any two real numbers. Not only does the commutative law of multiplication hold for these matrices (proof?), but other properties can be observed. We are going to let you experiment a little to see if some of the ordinary rules for a field are true.
Now suppose that a = 0 and b = 1. We shall denote the resulting matrix by J. That is,
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If we square this matrix, the result is
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Does this suggest anything? If b = 0 and a is any real number, we obtain the matrix
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which we have identified with the real number a. Thus, our new system of matrices includes all the real numbers, and also a number J that acts like i. Indeed, if we define i as the matrix J, then, as above, i2 = J2 = (–1)I = –1.
You might want to explore this system further on your own, and to continue the development. It provides the very number system we were looking for, one that contains the real number system in a consistent manner, one that obeys all the desirable algebraic rules, and one that shows how i fits in algebraically. We do not have to invoke artificial rules or add axioms to produce desired algebraic properties. All the familiar rules of algebra (the field properties) hold for these 2u2 matrices. It constitutes the field of complex numbers, denoted by C, a field that includes R as a subfield. (You are encouraged to work out the essential details of this system.)
GRAPHING COMPLEX NUMBERS—THE POLAR REPRESENTATION
An old name for a visual (geometric) treatment of complex numbers is the Argand diagram. Its construction was an attempt by mathematicians to explain complex numbers using geometry. The principal persons involved with this discovery (which was made independently at about the same time) were a Norwegian surveryor and cartographer Caspar Wessel (1745–1818), a French-Swiss bookkeeper Jean-Robert Argand (1768–1822), and Carl Friedrich Gauss (1777–1855); (Burton, 1991).
Since a complex number z = x + iy consists of a unique pair of real numbers (x and y), it can be plotted as a point in the xy-plane. Thus, for example, 2 + i, 3i, and –4 – 3i are represented by unique points, as shown in Figure 1.1. With this technique, a graphical representation of complex numbers emerges. The xy-plane may thus be referred to as the complex plane. The number 3i, for example (called pure imaginary), lies on the positive y-axis 3 units from the origin. When z is real (y = 0), it lies on the x-axis (called the real axis), and when z is pure imaginary (x = 0), it lies on the y-axis (called the imaginary axis); all other complex numbers are located in one of the four quadrants.
The arithmetic operations for complex numbers now take on a geometric meaning. Addition is actually vector addition, obeying the parallelogram law: If z = x + iy and w = u + iv, then z + w (x + u) + i(y + v). Thus the sum z + w is the fourth vertex of the parallelogram having consecutive vertices z, O (the origin), and w, as illustrated in Figure 1.2, just as is the case for vector addition.
Another property of addition comes from this geometric interpretation. The absolute value of a complex number z, denoted by |z|, is defined to be the distance from the origin to z (sometimes called the modulus or magnitude of z). That is, by the Pythagorean theorem, |z| = |x + iy|= [square root of (x2 + y2. One can then observe from Figure 1.2 the important triangle inequality for complex nmbers
(1.1) |z + w| ≤ |z| + |w|
In Figure 1.2, the length of the dashed line joining z and z + w equals the distance from O to w, or |w| (due to properties of parallelograms), so the geometric triangle inequality in the triangle having vertices O, z, and z + w proves (1.1).
Multiplication involves two geometric properties. First, the distance from O to the product zw is the product of the individual distances from O to z and from O to w. That is,
(1.2) |zw| = |z|x|w|
Secondly, the angle which the ray from O to zw makes with the positive x-axis is the sum of the two angles θ and φ made by Oz and Ow individually, as in Figure 1.3. That is,
(1.3) arg zw = arg z + arg w
where, in general, arg z denotes the argument of the complex number z, defined as the angle which the line Oz makes with the positive x-axis. This property is the first of many intriguing features of complex numbers to be revealed throughout this book.
Establishing (1.2) and (1.3) requires the use of polar coordinates. Recall this concept from algebra/trigonometry: each point P(x, y) in the plane except the origin can be represented by polar coordinates (r, θ), where r is the signed distance from O to P, and [thet] is the measure of the angle from the positive x-axis to the ray OP (Figure 1.4). The number r can be negative if one identifies (–r, θ) with the point (r, θ + π), which lies on the ray opposite ray OP; and θ can be any multiple of 2π plus the initial angle defined by ray OP. The following conversion formulas should be familiar to you, valid for all possible values of r and θ:
(1.4)x = r cos θ and y = r sin θ
Although r and θ can be any two real numbers, for definiteness we shall from now on restrict r to be positive and require θ to lie on the interval (–π, π]. If ( [x, y]) ≠ (0, 0), the formulas (1.4) may then be inverted to produce:
(1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where the signs and value of k are chosen so that –π θ ≤ π. For convenience, tan–1y/x is taken as a symbol for (1) π/2 if x = 0 and y > 0, and (2) –π/2 if x = 0 and y< 0. Thus, in general, θ = tan–1y/x if x ≥ 0, θ = tan–1y/x + π if x < 0 and y ≥ 0, and θ = tan–1y/x – π if x and y are both negative (θ is not defined if x and y are both zero. With this restriction, we define θ as the principle argument of the complex number z, denoted Argz.
Writing a complex number in polar form amounts to representing (x, y) in polar coordinates (r, θ), and substituting (1.4) into z. This produces the expression
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Note that the engineer's shorthand symbol cis θ for cos θ + i sin θ was used; this will occasionally appear in future work. The terms of the polar form of x + iy may be written entirely in terms of x and y (in the case of the general polar form, the answers are not uniquely determined):
(1.6) General Polar Form for z = x + iy: z = r cis where r2 = x2 + y2 and tan θ = y/x.
(1.7) Restricted Polar Form:z = r cis θ where r = [square root of (x2 + y2)] and θ = tan-1 [y/x] [+ or -] kr (asin (1.5))
NUMERICAL EXPERIMENT
Consider the complex numbers 3 + 4i and 12 – 5i. Write these numbers in polar form using 5-decimal accuracy for θ (and φ) as in Example 2(d). This leads to
3 + 4i = r cis θ and 12 – 5i = s cis φ
(r and s are integers.) Next, take the product (3 + 4i)(12 – 5i) using ordinary algebra, then convert the answer to polar form
(3 + 4i)(12 - 5i) = t cis ω
where ω is a 5-place decimal. Now test the product properties mentioned above to see if t = rs and ω = θ + φ.
(Continues...)
Excerpted from A Basic Course in Complex Variables by David C. Kay. Copyright © 2014 David C. Kay. Excerpted by permission of iUniverse.
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