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Unformatted text preview: CHAPTER 2 FUNCTIONS OF A COMPLEX VARIABLE 2.1 COMPLEX NUMBERS In the course of study of roots of algebraic equations and in particular the cubic equation, it has been found convenient to introduce the concept of a number whose square is equal to —1. By a well-established tradition, this number is denoted by i, and we write i2 = —— l and i = V —- 1. If we allow ito be multiplied by real numbers, we obtain the so-called imaginary numbers* of the form bi (where b is real). If the usual rules of multiplication are extended to imaginary numbers, then we must conclude that the products of imaginary numbers are real numbers; moreover, their squares are negative real numbers. For instance, (3i)(-4i) = (3)(-4)i2 = (—12)(-1) = (-51‘)2 = (—5)2i2 = —25. If imaginary numbers are adjoined to real numbers, we have a system within which we can perform multiplication and division (except by zero, of course). We say that such a system is closed under multiplication and division. However, our system is not closed under addition and subtraction.1‘, To eliminate this de- ficiency, so-called complex numbers are introduced. These are numbers which are most often written in the form a + bi (a, b = real numbers) and are assumed to obey appropriate algebraic rules. As will be shown below, the system of complex numbers is closed under addition, subtraction, multiplica- tion, and division plus the "extraction of roots” operation. In short, it has all the desirable algebraic characteristics and represents an extension of the real number system. The study of complex numbers is invaluable for every physicist because the description of physical laws is much more complicated without them. * Imaginary numbers are also called pure imaginary numbers to stress the distinction from the more general case of complex numbers. The name originated from the belief that imaginary numbers, as well as complex numbers, do not represent directly observable quantities in nature. While this point of view is now mostly abandoned, the original nomenclature still exists. T The system is not closed under the operation of extraction of the square root either; for example, x/i is neither real nor (pure) imaginary 44 2.2 BASIC ALGEBRA AND GEOMETRY or COMPLEX NUMBERS 45 2.2 BASIC ALGEBRA AND GEOMETRY OF COMPLEX NUMBERS Ifcomplex numbers are written in the usual form a + ib (or a + bi) then the usual algebraic operations with them are defined as follows. 1. Addition; (01+ib1)+(a2 +1172): (41+ 02)+i(b1+ b2)- 2. Multiplication: (01 + ibi) ' ((12 +1132): (aiaz - [91132) + [(ale + 02171)- The second rule is easy to follow if we recognize that the expressions a + ib are multiplied in the same manner as binomials, using the distributive and associative laws, and i2 is replaced by — 1. Complex numbers of the form a + [O are tacitly identified with real numbers since they obey the same algebraic rules and are generally indistinguishable from each other.* Complex numbers of the form 0 + ib are then (pure) imaginary numbers. It is customary to write simply a + i0 = a and 0 + ib = ib. Sub- traction of complex numbers can be defined as inverse addition so that if (‘11 + [131) — (02 + ibz) = x + i)” then a1+ib1= (x + iy) + (az + ibz) from which it follows thatT x = a1 —— a2 and y =tb1— b2. An alternative is to form the negative of a complex number, —(a + ib) = (—l)(a + ib) = (—l + iO)(a + ib) = —a — ib, 'and reduce the subtraction to addition. The rule for division can be similarly deduced by inverting the multiplication. A shortcut method is given by the following technique: a+ib a+ibc—id_(ac+bd)+i(bc—ad) c+id=c+idc—id_ c2+d2 _ac+bd .bC—ad 2 2 —__c2+d2+lc———_2+d2 (c +d #0). It is readily seen that the divisor can be any complex number except zero (namely the number 0 + [0, which is unique and is written simply 0). * In a more rigorous language, “the subset of complex numbers of the form a + [0 is isomorphic to the set of real numbers under the correspondence 0 + i0 <—> a.” T It is tacitly postulated that x1 + iyl = x2 + iy2 if and only if x1 = x2 and y1 = y2. 46 FUNCTIONS OF A COMPLEX VARIABLE 2.2 Remarks 1. The addition of complex numbers obeys the same rule as the addition of vectors in plane, provided a and b are identified with components of a vector. Note, however, that the multiplication of complex numbers differs from the formation of dot and cross products of vectors. 2. The use of the symbol i and the related binomial a + ib is conventional, but not indispensable. It is possible to define a complex number as a pair of real numbers, (a, b), obeying certain peculiar rules, e.g., the multiplication can be defined by (a19b1)(a2) b2) = (ala2 — blb23alb2 + 02b1)a and so on. It should be clear that the form a + ib is just a representation of a complex number. It is customary to represent complex numbers by points in the so-called com- plex plane, or Argand diagram (Fig. 2.1). If we denote the complex number x + iy by a single symbol 2 and write 2 = x + iy, then to each 2 there cor- responds a point in the complex plane with the abscissa x and the ordinate y. This idea also leads us to the trigonometric representation of a complex number: 2 = r(cosO+ isin6), where r = x/x2 + y2 and tan 0 = y/x. In this representation r is unique (positive square root) but 6 is not. A common convention is to demand that'l' lm Imaginary axis x Re ‘T < 9 S 7", Real axis along with the standard rule of quadrants, ' Figure 2-1 namely, 0 < O ify < 0. The following nomenclature and notation will be widely used: If 2: x+iy= r(coso+isin6) then x = Re 2 is the real part of z, y = 1m 2 is the imaginary part of z, r = |z| is the modulus of 2, also known as the magnitude or absolute value of z, ' 0 is the argument of 2, also called the polar angle or phase: The number x — iy is called the complex conjugate Of the number 2 = x + iy and vice versa. We shall denote it by 2*. We can say that z and 2* represent (on the complex plane) the reflections of each other with respect to the real axis. T Another commonly used convention is 0 S 0 < Zr. 1 A more precise name for 0 would be the “principal value of the argument of 2” (see p. 57). 2.2 BASIC ALGEBRA AND GEOMETRY or COMPLEX NUMBERS 47 Remarks 1. The quantity 22* are the same). 2. The quantity 2 + are the same). 3. The rules (21 + 22)* = 21‘ + 2'5 and (2122)“ = remembered. is always a nonnegative real number equal to lzl 2 or to |z"‘|2 (which 2* is always a real number, equal to 2 Re 2 or to 2 Re 2* (which 2"sz are evident and should be Figure 2.2 Because complex numbers obey the same addition rule that applies to vectors in a plane, they can be added graphically by the parallelogram rule (Fig. 2.2a). Conversely, vectors in a plane can be represented by complex numbers. The scalar product of two such vectors can be obtained by the rule (21-22) = Re (2122) = Re (2.2;). e vectors corresponding to complex num- where it is understood that 21 and 22 ar duct can be obtained in a similar bers 21 and 22 respectively. The vector pro fashion: [21 x 221= 1m (2122) = —1m (2123‘). Exercise. Verify the validity of the above rules for scalar and vector products. In the theory of complex variables, the expression lzl - 221 is often used. According to Fig. 2.2(b) this quantity (modulus of the complex number 21 — 22) is equal to the distance between the points 21 and 22 in the complex plane. It follows that the statement l2 - 201 < R (which often occurs in proofs of various theorems) means geometrically that point z is within the circle of radius R drawn around the point 20 as a center (i.e., z is in the R-neighborhood of 20; see p. 16). The following two inequalities are easily proved from geometrical considerations: 1- 121 + Z2| 5 [211+ 122|~ (A side of a triangle is less than or equal to the sum of the other two sides.) 7-- izi — Zle “21'" lzzli- (The difierence of two sides of a triangle is less than or equal to the third side.) 48 FUNCTIONS or A COMPLEX VARIABLE 2.3 Remark. It should be emphasized that inequalities can exist only among the moduli of complex numbers, not among the complex numbers themselves. A complex number cannot be greater or smaller than another complex number. Also, there are no positive or negative complex numbers. 2.3 DE MOIVRE FORMULA AND THE CALCULATION OF ROOTS While addition and subtraction of complex numbers are most easily performed in their cartesian form 2 = x + iy, multiplication and division are easier in trigonometric form. Ile = r1(cos 01 + isin 01) and 22 = r2(cos 92 + [sin 02), then elementary calculation shows that 2122 = "1’2lC05 (91 + 02)+isin(01+ 92)] with the provision that if 01 + 02 happens to be greater than 1r, or less than or equal to —1r, then the amount 21r should be added or subtracted to fulfill the condition -—1r < (01 + 02) 5 1r. Remark. It should be emphasized that even though cos (0 :1: 21r) = cos 0 and sin (0 5; Zn) = sin 6, the value of 0 is supposed to be uniquely specified. This will be- come evident when 0 is subjected to certain operations, e. g., in the course of evaluation of roots. The convention -1r < 0 < 1r is not the only one possible, but some convention must be adopted and ours is just as good as any other Using the same trigonometric identities as in the above multiplication rule, we can also obtain the so-called De Moivre formula: (cos 0 + isin 0)" = cos n6 + isin n6 (n = integer). Thus we now have the general rule for calculating the nth power ofa complex num- ber z. Ifz = r(cos 6 + [sin 0), then 2” = R(cos ¢ + i sin ¢), where R = r" and ¢ = n6 i 27rk with the integer k chosen in such a way that —1r < (1) 3 1r. The rule for calculating the nth root of a complex number can now be derived without much difi’iculty. If 2 = r(cos 6 + isin 9), then the complex number We = {7; (cos; + isin Q) n is definitely the nth root of 2 because w}; = z. However,this is not the only nth root of z; the numbers W], = {l/r<cos—~—— 0+ 27rk + i sin 0+ —2—7rk) n n where k =1, 2, 3, . . . , (n — l), are also nth roots of 2 because w = 2. It is Figure 2.3 2.4 COMPLEX FUNCTIONS. EULER’S FORMULA 49 customary to call the number wo the principal root of z. The nth roots of a complex number 2 are always located _at the vertices of a regular polygon of n sides inscribed in a circle of radius R = {7 r about the origin (Fig. 2.3). Exercise. Verify that all possible roots of a complex number 2 are given by the above formulas. Show that all complex numbers except one have exactly n (different) nth-order roots. Which complex number is the exception? 2.4 COMPLEX FUNCTIONS. EULER’S FORMULA Complex numbers 2 = x + iy may be considered as variables if x or y (or both) vary. If this is so, then complex functions may be formed. For instance, consider the equation w = 22. If we write 2 = x + iy and w = u + iv, it follows that u=x—y, v=2xy. From this we conclude that if w is a function of 2, then u and v are, in general, functions of both x and y. Thus we are dealing with two (independent) real functions of two (independent) real variables. lm .\ o:=l+i Invariant point Re :=1 Figure 2.4 Figure 2.5 Graphical representation of complex functions poses a problem since we must deal with four real variables simultaneously. The idea of mapping is most com- monly used. Two separate complex planes, the z-plane and the w-plane, are considered side by side, and a point 20 is said to be mapped ont the point We = f(zo). For instance, formula w = 22 maps 21 ’= i onto wl = i2 = ——1; it also maps 22 = l + ionto W2 = 2i, 23 = 1 onto w3 = l, and so on. This is illustrated in Fig. 2.4, where it is also indicated that the horizontal line y = l in the z-plane is mapped onto the parabola u = 2\/u + l in the w-plane. Some- times it is convenient to superimpose the two planes. Then the images of various points are located on the same plane and the function w = f(z) is said to trans- form the complex plane into itself (or a part of itself), as in Fig. 2.5, for the same function w = 22. 50 FUNCTIONS or A COMPLEX VARIABLE 24 Exercise. Show that the function w = iz represents counterclockwise rotation of the complex plane by 90°. How would you describe a rotation by 180°? How would you describe a clockwise rotation by 90° ? Algebraic functions of a complex variable are defined by algebraic operations which are directly applicable to complex numbers. Transcendental functions, however, may require special definitions. Consider, for instance, the exponential function e‘ (real x). Its basic properties are 1. em”2 = exle”, 2. (e2)“ = e” It is desired to define a com lex ex onential function e2 with the same ro er- P P P P ties. Write z = x + I'y; then e2 = ez+iy = e’ei”. The quantity e is a well- defined real number, but how shall we define e1”? One possible method is as follows. Assume that e‘” can be represented by the usual power series 2 3 =1+(iy)+(—2——ly,) +(——'y,) + Then, rearranging the terms, we have 2 y4 3 5 ew‘<1‘_+l—!“”)+i< _§_! %_...) =cosy+isiny. The validity of this procedure can be established after the development of the theory of convergence for complex series. However, at this stage we may simply define the function e‘” by means of e” = cosy + isin y. This is Euler’s formula. The desired properties, ei‘ylfl’z) = emei“, (eiy)" = emy , (n = integer), follow from the identities (cosy1+isiny1)(cosy2 + isinyz) = 0050/1 + M) + isin (y; + M) d . . . . an (cosy + Ism y)" = cos ny + zsm ny. The definition of a complex exponential function is then given by the formula e2 = e‘(cosy + isin y) which has the desired properties and reduces to the real exponential function if Im z = O. 2.5 APPLICATIONS OF EULER’S FORMULA 51 2.5 APPLICATIONS OF EULER’S FORMULA Euler’s formula leads to the compact polar representation of complex numbers, z= x+iy= r(cosB+isin0)= ref”. Suppose that a complex number 2 is multiplied by ei", where a is a real con- stant. Then The new number can be obtained by rotating the point 2 about the origin by an angle a. This fact has many important applications. Euler’s formula also permits the description of sinusoidally varying real quantities by means of complex exponentials. A general form of such quantity is f(t) = acos (wt — 0), where a (amplitude), to (angular frequency), and 0 (phase) are constants, and t is a real variable (usually time). Consider the complex function of the real' variable g(t) where B is a complex constant. Set B = aew; then II W m l 2 g(t) = ae‘ee’iw’ = acos (0 — wt) + ia sin (0 — wt) = acos (wt — 0) — ia sin (wt — 6). In other words, f (t) = Re {g(t)}. Complex functions of a real variable can of real variables. For instance, if g0) = “(0 + 1'00) (11,0 be treated by the methods of calculus real functions), then ' [email protected] .__’ E’d:+’dz and so on. Differentiation of Be’i‘”l is very simple: fl (Be—M) = —z'wBe dt —1"w t The use of complex exponentials is illustrated in the following example. Consider a (damped) harmonic oscillator subject to a harmonically varying external force. The differential equation to be solved reads 5: + 2ax + 02%): = Fcos (wt — ¢) (x = (dx/dl) etc.), Where the constants a, coo, F, to, and ¢ are real, and both variables x and t are real. 34 rUNCHUNs or A COMPLEX VARIABLE 2.6 2.6 MULTIVALUED FUNCTIONS AND RIEMANN SURFACES Certain complex functions are multivalued and they are usually considered as consisting of branches, each branch being a single-valued function of 2. For instance, f (z) = \/E can be split into two branches according to the usual formula for the roots (2 = re”): 1. Principal branch, f1(2) = v; em“), 2. Second branch, f2(z) = \fl eil<9+21r)/21_ Strictly speaking, f1(z) and f 2(2) are two separate functions but they are intimately connected and for this reason they are treated together as two branches of a (double-valued) function f (z) = x/E. Note that the principal branch does not map the z-plane onto the entire w-plane, but rather onto the right half-plane (Re w > 0) to which the positive imaginary semiaxis is added. The negative imaginary semiaxis is not included. The second branch, which has no special name, maps the z-plane onto the left half-plane (Re w < 0) plus the negative imaginary semiaxis. Except for z = 0, no other point on the w-plane (image plane) is duplicated by both mappings. Also observe another important feature of the two branches. Each branch taken separately is discontinuous on the negative real semiaxis. The meaning of this is as follows: The points i(1r—5) 21 = e t(—1r+6), and 22 = e where 6 is a small positive number, are very close to each other. However, their images under the principal branch mapping, namely f1(21) = awn—M2) and f1(22) = e—“flz—é/m, are very far from each other. On the other hand, note that the image of 22 under the mapping/2(2), namely, f2(22) ___ ei(1r/2+6/2)’ is very close to the pointf1(zl). It appears that the continuity of mapping can be preserved if we switch branches as we cross the negative real semiaxis. To give this idea a more precise meaning we must define the concept of con- tinuous function of a complex variable. Let w = f(z) be defined in some neigh- borhood (see pp. 47 and 16) of point 20 and let f(zo) = W5. We say that f(z) is continuous at 20 if* f(z) —> wo whenever z —> 20 in the sense that given 6 > 0 (arbitrarily small), the inequality lf(z) — wol < 6 holds whenever [z -— 20] < 6 holds, for sufficiently small 6. It is readily shown'l' that if w = u(x, y) + iv(x, y), then the continuity of w implies the continuity of u(x, y) and v(x, y) and vice versa. * Also written as limb“0 f (z) = f(zo). T For example, see Kaplan, p. 495. joir Two edges joined here ; 2.6 MULTIVALUED FUNCTIONS AND RIEMANN SURFACES 55 Riemann proposed an ingenious device to represent both branches by means of a single continuous mapping: Imagine two separate z-planes cut along the negative real semiaxis from “minus infinity” to zero. Imagine that the planes are superimposed on each other but retain their separate identity in the manner of two sheets of paper laid on top of each other. Now suppose that the second quadrant of the upper sheet is joined along the cut to the fourth quadrant of the lower sheet to form a continuous surface (Fig. 2.6). It is now possible to start a curve C in the third quadrant of the upper sheet, go around the origin, and cross the negative real semiaxis into the third quadrant of the lower sheet in a con- tinuous motion (remaining on the surface). The curve can be continued on the lower sheet around the origin into the second quadrant of the lower sheet. Lower sheet Figure 2.6 Figure 2.7 Now imagine the second quadrant of the lower sheet joined to the third quadrant of the upper sheet along the same cut (independently of the first joint and actually disregarding its existence). The curve C can then be continued onto the upper sheet and may return to the starting point. This process of cutting and cross- joining two planes leads to the formation of a Riemann surface which is thought of as a single continuous surface formed of two Riemann sheets (Fig. 2.7). An important remark is now in order: The line between the second quadrant of the upper sheet and the third quadrant of the lower sheet is to be considered as distinct from the line between the second quadrant of the lower sheet and the third quadrant of the upper one. This is where the paper model fails us. According to this model the negative real semiaxis appears as the line where all four edges of our cuts meet. However, the Riemann surface has no such property; there are two real negative semiaxes on the Riemann surface just as there are two real posi- tive semiaxes. The mapping [(2) = \/3 may help to visualize this: The principal branch maps the upper Riemann sheet (negative real semiaxis excluded) onto the region Re w > O of the w-plane. The line joining the second upper with the third lower quadrants is also mapped by the principal branch onto the positive imaginary semiaxis. The lower Ri...
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