Unformatted text preview: Chapter 1 Complex Numbers
Die ganzen Zahlen hat der liebe Gott geschaffen, alles andere ist Menschenwerk. (God created the integers, everything else is made by humans.) Leopold Kronecker (18231891) 1.0 Introduction The real numbers have nice properties. There are operations such as addition, subtraction, multiplication as well as division by any real number except zero. There are useful laws that govern these operations such as the commutative and distributive laws. You can also take limits and do calculus. But you cannot take the square root of 1. Equivalently, you cannot find a root of the equation x2 + 1 = 0. (1.1) Most of you have heard that there is a "new" number i that is a root of the Equation (1.1). That is, i2 + 1 = 0 or i2 = 1. We will show that when the real numbers are enlarged to a new system called the complex numbers that includes i, not only do we gain a number with interesting properties, but we do not lose any of the nice properties that we had before. Specifically, the complex numbers, like the real numbers, will have the operations of addition, subtraction, multiplication as well as division by any complex number except zero. These operations will follow all the laws that we are used to such as the commutative and distributive laws. We will also be able to take limits and do calculus. And, there will be a root of Equation (1.1). In the next section we show exactly how the complex numbers are set up and in the rest of this chapter we will explore the properties of the complex numbers. These properties will be both algebraic properties (such as the commutative and distributive properties mentioned already) and also geometric properties. You will see, for example, that multiplication can be described geometrically. In the rest of the book, the calculus of complex numbers will be built on the propeties that we develop in this chapter. 1.1 Definition and Algebraic Properties
C = {(x, y) : x, y R} , 1 The complex numbers can be defined as pairs of real numbers, CHAPTER 1. COMPLEX NUMBERS equipped with the addition (x, y) + (a, b) = (x + a, y + b) and the multiplication (x, y) (a, b) = (xa  yb, xb + ya) . 2 One reason to believe that the definitions of these binary operations are "good" is that C is an extension of R, in the sense that the complex numbers of the form (x, 0) behave just like real numbers; that is, (x, 0) + (y, 0) = (x + y, 0) and (x, 0) (y, 0) = (x y, 0). So we can think of the real numbers being embedded in C as those complex numbers whose second coordinate is zero. The following basic theorem states the algebraic structure that we established with our definitions. Its proof is straightforward but nevertheless a good exercise. Theorem 1.1. (C, +, ) is a field; that is: (x, y), (a, b) C : (x, y) + (a, b) C (x, y), (a, b), (c, d) C : (x, y) + (a, b) + (c, d) = (x, y) + (a, b) + (c, d) (x, y), (a, b) C : (x, y) + (a, b) = (a, b) + (x, y) (x, y) C : (x, y) + (0, 0) = (x, y) (x, y) C : (x, y) + (x, y) = (0, 0) (x, y), (a, b) C : (x, y) (a, b) C (x, y), (a, b), (c, d) C : (x, y) (a, b) (c, d) = (x, y) (a, b) (c, d) (x, y), (a, b) C : (x, y) (a, b) = (a, b) (x, y) (x, y) C : (x, y) (1, 0) = (x, y) x (x, y) C \ {(0, 0)} : (x, y) x2 +y2 , x2y 2 = (1, 0) +y (1.2) (1.3) (1.4) (1.5) (1.6) (1.7) (1.8) (1.9) (1.10) (1.11) Remark. What we are stating here can be compressed in the language of algebra: equations (1.2) (1.6) say that (C, +) is an Abelian group with unit element (0, 0), equations (1.7)(1.11) that (C \ {(0, 0)}, ) is an abelian group with unit element (1, 0). (If you don't know what these terms meandon't worry, we will not have to deal with them.) The definition of our multiplication implies the innocent looking statement (0, 1) (0, 1) = (1, 0) . This identity together with the fact that (a, 0) (x, y) = (ax, ay) allows an alternative notation for complex numbers. The latter implies that we can write (x, y) = (x, 0) + (0, y) = (x, 0) (1, 0) + (y, 0) (0, 1) . If we thinkin the spirit of our remark on the embedding of R in Cof (x, 0) and (y, 0) as the real numbers x and y, then this means that we can write any complex number (x, y) as a linear (1.12) CHAPTER 1. COMPLEX NUMBERS 3 combination of (1, 0) and (0, 1), with the real coefficients x and y. (1, 0), in turn, can be thought of as the real number 1. So if we give (0, 1) a special name, say i, then the complex number that we used to call (x, y) can be written as x 1 + y i, or in short, x + iy . The number x is called the real part and y the imaginary part1 of the complex number x + iy, often denoted as Re(x + iy) = x and Im(x + iy) = y. The identity (1.12) then reads i2 = 1 . We invite the reader to check that the definitions of our binary operations and Theorem 1.1 are coherent with the usual real arithmetic rules if we think of complex numbers as given in the form x + iy. 1.2 Geometric Properties Although we just introduced a new way of writing complex numbers, let's for a moment return to the (x, y)notation. It suggests that one can think of a complex number as a twodimensional real vector. When plotting these vectors in the plane R2 , we will call the xaxis the real axis and the yaxis the imaginary axis. The addition that we defined for complex numbers resembles vector addition. The analogy stops at multiplication: there is no "usual" multiplication of two vectors that gives another vectormuch less so if we additionally demand our definition of the product of two complex numbers. z1 z2 z1 + z2 Figure 1.1: Addition of complex numbers. Any vector in R2 is defined by its two coordinates. On the other hand, it is also determined by its length and the angle it encloses with, say, the positive real axis; let's define these concepts thoroughly. The absolute value (sometimes also called the modulus) of x + iy is r = x + iy = x2 + y 2 , and an argument of x + iy is a number such that x = r cos 1 and y = r sin . The name has historical reasons: people thought of complex numbers as unreal, imagined. CHAPTER 1. COMPLEX NUMBERS 4 This means, naturally, that any complex number has many arguments; more precisely, all of them differ by a multiple of 2. The absolute value of the difference of two vectors has a nice geometric interpretation: it is the distance of the (end points of the) two vectors (see Figure 1.2). It is very useful to keep this geometric interpretation in mind when thinking about the absolute value of the difference of two complex numbers. z1 z1  z2 z2 Figure 1.2: Geometry behind the "distance" between two complex numbers. The first hint that absolute value and argument of a complex number are useful concepts is the fact that they allow us to give a geometric interpretation for the multiplication of two complex numbers. Let's say we have two complex numbers, x1 + iy1 with absolute value r1 and argument 1 , and x2 + iy2 with absolute value r2 and argument 2 . This means, we can write x1 + iy1 = (r1 cos 1 ) + i(r1 sin 1 ) and x2 + iy2 = (r2 cos 2 ) + i(r2 sin 2 ) To compute the product, we make use of some classic trigonometric identities: (x1 + iy1 )(x2 + iy2 ) = (r1 cos 1 ) + i(r1 sin 1 ) (r2 cos 2 ) + i(r2 sin 2 ) So the absolute value of the product is r1 r2 and (one of) its argument is 1 + 2 . Geometrically, we are multiplying the lengths of the two vectors representing our two complex numbers, and adding their angles measured with respect to the positive xaxis.2 In view of the above calculation, it should come as no surprise that we will have to deal with quantities of the form cos + i sin (where is some real number) quite a bit. To save space, bytes, ink, etc., (and because "Mathematics is for lazy people"3 ) we introduce a shortcut notation and define ei = cos + i sin . At this point, this exponential notation is indeed purely a notation. We will later see that it has an intimate connection to the complex exponential function. For now, we motivate this maybe strangeseeming definition by collecting some of its properties. The reader is encouraged to prove them.
One should convince oneself that there is no problem with the fact that there are many possible arguments for complex numbers, as both cosine and sine are periodic functions with period 2. 3 Peter Hilton (Invited address, Hudson River Undergraduate Mathematics Conference 2000)
2 = (r1 r2 cos 1 cos 2  r1 r2 sin 1 sin 2 ) + i(r1 r2 cos 1 sin 2 + r1 r2 sin 1 cos 2 ) = r1 r2 (cos 1 cos 2  sin 1 sin 2 ) + i(cos 1 sin 2 + sin 1 cos 2 ) = r1 r2 cos(1 + 2 ) + i sin(1 + 2 ) . CHAPTER 1. COMPLEX NUMBERS ............... .... 1 + 2....... . ... ... 2 ... ... .. . . . . . . . . .. . ... .. .. z ..... . 2 .. z1 . .. . .. . . .. ... .. 1 . . . . . . . . . . . . . . . . . .. .. . . 5 z1 z2 Figure 1.3: Multiplication of complex numbers. Lemma 1.2. For any , 1 , 2 R, (a) ei1 ei2 = ei(1 +2 ) (b) 1/ei = ei (c) ei(+2) = ei (d) ei = 1 (e)
d d ei = i ei . With this notation, the sentence "The complex number x+iy has absolute value r and argument " now becomes the identity x + iy = rei . The lefthand side is often called the rectangular form, the righthand side the polar form of this complex number. From very basic geometric properties of triangles, we get the inequalities z Re z z and  z Im z z . (1.13) The square of the absolute value has the nice property x + iy2 = x2 + y 2 = (x + iy)(x  iy) . This is one of many reasons to give the process of passing from x + iy to x  iy a special name: x  iy is called the (complex) conjugate of x + iy. We denote the conjugate by x + iy = x  iy . Geometrically, conjugating z means reflecting the vector corresponding to z with respect to the real axis. The following collects some basic properties of the conjugate. Their easy proofs are left for the exercises. Lemma 1.3. For any z, z1 , z2 C, (a) z1 z2 = z1 z2 CHAPTER 1. COMPLEX NUMBERS (b) z1 z2 = z1 z2 (c) z1 = z1 z2 z2 (d) z = z (e) z = z (f) z2 = zz (g) Re z = (h) Im z =
1 2 6 (z + z) (z  z) 1 2i (i) ei = ei . From part (f) we have a neat formula for the inverse of a nonzero complex number: z 1 = 1 z = 2. z z A famous geometric inequality (which holds for vectors in Rn ) is the triangle inequality z1 + z2  z1  + z2  . By drawing a picture in the complex plane, you should be able to come up with a geometric proof of this inequality. To prove it algebraically, we make extensive use of Lemma 1.3: z1 + z2 2 = (z1 + z2 ) (z1 + z2 ) = (z1 + z2 ) (z1 + z2 ) = z 1 z1 + z1 z 2 + z 2 z1 + z2 z 2 = z1 2 + z1 z2 + z1 z2 + z2 2 Finally by (1.13) z1 + z2 2 z1 2 + 2 z1 z2  + z2 2 = z1 2 + 2 Re (z1 z2 ) + z2 2 . = z1 2 + 2 z1  z2  + z2 2 = z1 2 + 2 z1  z2  + z2 2 = (z1  + z2 )2 , which is equivalent to our claim. For future reference we list several variants of the triangle inequality: Lemma 1.4. For z1 , z2 , C, we have the following identities: (a) The triangle inequality: z1 z2  z1  + z2 . CHAPTER 1. COMPLEX NUMBERS (b) The reverse triangle inequality: z1 z2  z1   z2 . n n (c) The triangle inequality for sums: zk zk . k=1 k=1 7 The first inequality is just a rewrite of the original triangle inequality, using the fact that z = z, and the last follows by induction. The reverse triangle inequality is proved in Exercise 15. 1.3 Elementary Topology of the Plane In Section 1.2 we saw that the complex numbers C, which were initially defined algebraically, can be identified with the points in the Euclidean plane R2 . In this section we collect some definitions and results concerning the topology of the plane. While the definitions are essential and will be used frequently, we will need the following theorems only at a limited number of places in the remainder of the book; the reader who is willing to accept the topological arguments in later proofs on faith may skip the theorems in this section. Recall that if z, w C, then z  w is the distance between z and w as points in the plane. So if we fix a complex number a and a positive real number r then the set of z satisfying z  a = r is the set of points at distance r from a; that is, this is the circle with center a and radius r. The inside of this circle is called the open disk with center a and radius r, and is written Dr (a). That is, Dr (a) = {z C : z  a < r}. Notice that this does not include the circle itself. We need some terminology for talking about subsets of C. Definition 1.5. Suppose E is any subset of C. (a) A point a is an interior point of E if some open disk with center a lies in E. (b) A point b is a boundary point of E if every open disk centered at b contains a point in E and also a point that is not in E. (c) A point c is an accumulation point of E if every open disk centered at c contains a point of E different from c. (d) A point d is an isolated point of E if it lies in E and some open disk centered at d contains no point of E other than d. The idea is that if you don't move too far from an interior point of E then you remain in E; but at a boundary point you can make an arbitrarily small move and get to a point inside E and you can also make an arbitrarily small move and get to a point outside E. Definition 1.6. A set is open if all its points are interior points. A set is closed if it contains all its boundary points. Example 1.7. For R > 0 and z0 C, {z C : z  z0  < R} and {z C : z  z0  > R} are open. {z C : z  z0  R} is closed. Example 1.8. C and the empty set are open. They are also closed! CHAPTER 1. COMPLEX NUMBERS 8 Definition 1.9. The boundary of a set E, written E, is the set of all boundary points of E. The interior of E is the set of all interior points of E. The closure of E, written E, is the set of points in E together with all boundary points of E. Example 1.10. If G is the open disk {z C : z  z0  < R} then G = {z C : z  z0  R} That is, G is a closed disk and G is a circle. One notion that is somewhat subtle in the complex domain is the idea of connectedness. Intuitively, a set is connected if it is "in one piece." In the reals a set is connected if and only if it is an interval, so there is little reason to discuss the matter. However, in the plane there is a vast variety of connected subsets, so a definition is necessary. Definition 1.11. Two sets X, Y C are separated if there are disjoint open sets A and B so that X A and Y B. A set W C is connected if it is impossible to find two separated nonempty sets whose union is equal to W . A region is a connected open set. The idea of separation is that the two open sets A and B ensure that X and Y cannot just "stick together." It is usually easy to check that a set is not connected. For example, the intervals X = [0, 1) and Y = (1, 2] on the real axis are separated: There are infinitely many choices for A and B that work; one choice is A = D1 (0) (the open disk with center 0 and radius 1) and B = D1 (2) (the open disk with center 2 and radius 1). Hence their union, which is [0, 2] \ {1}, is not connected. On the other hand, it is hard to use the definition to show that a set is connected, since we have to rule out any possible separation. One type of connected set that we will use frequently is a curve. Definition 1.12. A path or curve in C is the image of a continuous function : [a, b] C, where [a, b] is a closed interval in R. The path is smooth if is differentiable. We say that the curve is parametrized by . It is a customary and practical abuse of notation to use the same letter for the curve and its parametrization. We emphasize that a curve must have a parametrization, and that the parametrization must be defined and continuous on a closed and bounded interval [a, b]. Since we may regard C as identified with R2 , a path can be specified by giving two continuous realvalued functions of a real variable, x(t) and y(t), and setting (t) = x(t) + y(t)i. A curve is closed if (a) = (b) and is a simple closed curve if (s) = (t) implies s = a and t = b or s = b and t = a, that is, the curve does not cross itself. The following seems intuitively clear, but its proof requires more preparation in topology: Proposition 1.13. Any curve is connected. The next theorem gives an easy way to check whether an open set is connected, and also gives a very useful property of open connected sets. Theorem 1.14. If W is a subset of C that has the property that any two points in W can be connected by a curve in W then W is connected. On the other hand, if G is a connected open subset of C then any two points of G may be connected by a curve in G; in fact, we can connect any two points of G by a chain of horizontal and vertical segments lying in G. and G = {z C : z  z0  = R} . CHAPTER 1. COMPLEX NUMBERS 9 A chain of segments in G means the following: there are points z0 , z1 , . . . , zn so that, for each k, zk and zk+1 are the endpoints of a horizontal or vertical segment which lies entirely in G. (It is not hard to parametrize such a chain, so it determines is a curve.) As an example, let G be the open disk with center 0 and radius 2. Then any two points in G can be connected by a chain of at most 2 segments in G, so G is connected. Now let G0 = G \ {0}; this is the punctured disk obtained by removing the center from G. Then G is open and it is connected, but now you may need more than two segments to connect points. For example, you need three segments to connect 1 to 1 since you cannot go through 0. Warning: The second part of Theorem 1.14 is not generally true if G is not open. For example, circles are connected but there is no way to connect two distinct points of a circle by a chain of segments which are subsets of the circle. A more extreme example, discussed in topology texts, is the "topologist's sine curve," which is a connected set S C that contains points that cannot be connected by a curve of any sort inside S. The reader may skip the following proof. It is included to illustrate some common techniques in dealing with connected sets. Proof of Theorem 1.14. Suppose, first, that any two points of G may be connected by a path that lies in G. If G is not connected then we can write it as a union of two nonempty separated subsets X and Y . So there are disjoint open sets A and B so that X A and Y B. Since X and Y are disjoint we can find a X and b G. Let be a path in G that connects a to b. Then X = X and Y = Y are disjoint and nonempty, their union is , and they are separated by A and B. But this means that is not connected, and this contradicts Proposition 1.13. Now suppose that G is a connected open set. Choose a point z0 G and define two sets: A is the set of all points a so that there is a chain of segments in G connecting z0 to a, and B is the set of points in G that are not in A. Suppose a is in A. Since a G there is an open disk D with center a that is contained in G. We can connect z0 to any point z in D by following a chain of segments from z0 to a, and then adding at most two segments in D that connect a to z. That is, each point of D is in A, so we have shown that A is open. Now suppose b is in B. Since b G there is an open disk D centered at b that lies in G. If z0 could be connected to any point in D by a chain of segments in G then, extending this chain by at most two more segments, we could connect z0 to b, and this is impossible. Hence z0 cannot connect to any point of D by a chain of segments in G, so D B. So we have shown that B is open. Now G is the disjoint union of the two open sets A and B. If these are both nonempty then they form a separation of G, which is impossible. But z0 is in A so A is not empty, and so B must be empty. That is, G = A, so z0 can be connected to any point of G by a sequence of segments in G. Since z0 could be any point in G, this finishes the proof. 1.4 Theorems from Calculus Here are a few theorems from real calculus that we will make use of in the course of the text. Theorem 1.15 (ExtremeValue Theorem). Any continuous realvalued function defined on a closed and bounded subset of Rn has a minimum value and a maximum value. CHAPTER 1. COMPLEX NUMBERS 10 Theorem 1.16 (MeanValue Theorem). Suppose I R is an interval, f : I R is differentiable, and x, x + x I. Then there is 0 < a < 1 such that f (x + x)  f (x) = f (x + ax) . x Many of the most important results of analysis concern combinations of limit operations. The most important of all calculus theorems combines differentiation and integration (in two ways): Theorem 1.17 (Fundamental Theorem of Calculus). Suppose f : [a, b] R is continuous. Then x (a) If F is defined by F (x) = a f (t) dt then F is differentiable and F (x) = f (x). b (b) If F is any antiderivative of f (that is, F = f ) then a f (x) dx = F (b)  F (a).
2 2 For functions of several variables we can perform differentiation operations, or integration operations, in any order, if we have sufficient continuity: f f Theorem 1.18 (Equality of mixed partials). If the mixed partials xy and yx are defined on an open set G and are continuous at a point (x0 , y0 ) in G then they are equal at (x0 , y0 ). Theorem 1.19 (Equality of iterated integrals). If f is continuous on the rectangle given by a bd db x b and c y d then the iterated integrals a c f (x, y) dy dx and c a f (x, y) dx dy are equal. Finally, we can apply differentiation and integration with respect to different variables in either order: Theorem 1.20 (Leibniz's4 Rule). Suppose f is continuous on the rectangle R given by a x b and c y d, and suppose the partial derivative f exists and is continuous on R. Then x d dx d f (x, y) dy = c d c f (x, y) dy . x Exercises
1. Find the real and imaginary parts of each of the following: (a) (b) (c)
za z+a 3+5i 7i+1 . (a R). (d) in for any n Z. 2. Find the absolute value and conjugate of each of the following: (b) (2 + i)(4 + 3i).
4 Named after Gottfried Wilhelm Leibniz (16461716). For more information about Leibnitz, see http://wwwgroups.dcs.stand.ac.uk/history/Biographies/Leibnitz.html. 3 1+i 3 . 2 (a) 2 + i. CHAPTER 1. COMPLEX NUMBERS (c)
3i . 2+3i 11 (d) (1 + i)6 . 3. Write in polar form: (a) 2i. (b) 1 + i. (c) 3 + 3i. 4. Write in rectangular form: (a) 2 ei3/4 . (b) 34 ei/2 . (c) ei250 . 5. Find all solutions to the following equations: (a) z 6 = 1. (b) z 4 = 16. (d) z 6  z 3  2 = 0. 6. Show that (a) z is a real number if and only if z = z; (b) z is either real or purely imaginary if and only if (z)2 = z 2 . 7. Find all solutions of the equation z 2 + 2z + (1  i) = 0. 8. Prove Theorem 1.1. 9. Show that if z1 z2 = 0 then z1 = 0 or z2 = 0. 10. Prove Lemma 1.2. 11. Use Lemma 1.2 to derive the triple angle formulas: (b) sin 3 = 3 cos2 sin  sin3 . 12. Prove Lemma 1.3. 13. Sketch the following sets in the complex plane: (b) {z C : z  1 + i 2} . (a) {z C : z  1 + i = 2} . (c) {z C : Re(z + 2  2i) = 3} . (a) cos 3 = cos3  3 cos sin2 . (c) z 6 = 9. CHAPTER 1. COMPLEX NUMBERS (d) {z C : z  i + z + i = 3} . 14. Suppose p is a polynomial with real coefficients. Prove that (a) p(z) = p (z). (b) p(z) = 0 if and only if p (z) = 0. 15. Prove the reverse triangle inequality z1  z2  z1   z2 . 16. Use the previous exercise to show that z 21 1 for every z on the circle z = 2ei . 3 1 (a) z + 3 < 2. (c) 0 < z  1 < 2. 12 17. Sketch the following sets and determine whether they are open, closed, or neither; bounded; connected. (b) Im z < 1. (d) z  1 + z + 1 = 2. (e) z  1 + z + 1 < 3. 18. What are the boundaries of the sets in the previous exercise? 19. The set E is the set of points z in C satisfying either z is real and 2 < z < 1, or z < 1, or z = 1 or z = 2. (a) Sketch the set E, being careful to indicate exactly the points that are in E. (b) Determine the interior points of E. (c) Determine the boundary points of E. (d) Determine the isolated points of E. 20. The set E in the previous exercise can be written in three different ways as the union of two disjoint nonempty separated subsets. Describe them, and in each case say briefly why the subsets are separated. 21. Let G be the annulus determined by the conditions 2 < z < 3. This is a connected open set. Find the maximum number of horizontal and vertical segments in G needed to connect two points of G. d 22. Prove Leibniz's Rule: Define F (x) = c f (x, y) dy, get an expression for F (x)  F (a) as an iterated integral by writing f (x, y)  f (a, y) as the integral of f , interchange the order of x integrations, and then differentiate using the Fundamental Theorem of Calculus. ...
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 Topology, Real Numbers, Integers, Complex Numbers, Metric space, Complex number

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