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s0001 - WC APPENDIX Functions of a Complex Variable—...

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Unformatted text preview: WC APPENDIX Functions of a Complex Variable— Summary of Important Definitions and Theorems To cover complex variable theory in any detail in an appendix of the length of this one is impossible. Therefore, only the definitions and theorems important to the material in this text are summarized with few proofs given. Let s = a + jw denote a complex variable. Another complex variable W = U + jV is said to be a function of the complex variable s if, to each value of s in some set, there corresponds a value, or a set of values, of W. We denote the rule of correspondence between s and Why writing W = F (s). If for each value of s there is only one value of W, F (s) is said to be a singlewalued function of s. If more than one value of W corresponds to some value or set of values of s, F(s) is multivalued. EXAMPLE C-l (a) The function W = F (s) = s2 is single—valued. We can express U and Vin terms of (r and w as W=U+jV=(0'+jw)2 = (72 - w2 +j2crw (01) or U = 0'2 — (1)2 (C-Za) and V = 20w (C—2b) For each value of s = a + jw there corresponds one, and only one, value for W = U + jV. (b) The function W = F (s) = V3 is double-valued. To show this, we write s in polar form as s = reN’ (C-3) so that w = (ref¢)'/2 (C-4) Now e14) = ef<¢+2k”), where k = 0, 1. Therefore, W = rl/Zej(¢/2+k7r)’ k = 0’ 1 (05) and for each value of s there correspond two values of W, which are W1 2 r1/2ej¢/2 (C-6a) 566 App. 0 / Functions of a Complex Variable—Summary of Important Definitions and Theorems 567 and W2 2 r1/2ej(¢l2+7r) (C—6b) where r“2 is the positive square root of r. For example, with s = —4, these two values of W are W1 = 263W2 = 2j and W2 = 2ei("/2+") = —2j (c) W = s*, where the asterisk denotes the complex conjugate, is a single-valued function of s. If s = 0'+jw,thenW=F(s) = a—jw. We now state a few definitions pertaining to functions of a complex variable. DEFINITION 1: The neighborhood of a point so in the complex plane is the open circular disk is — sol < p where p > 0 is a real arbitrary constant. DEFINITION 2: A function F (s) is said to have the limit L as s approaches so if F (s) is defined in a neigh- borhood of so (except perhaps at so) and if, for every positive real number 6 > 0, we can find a real num- ber 8 such that [F(s) — L[ < e (07) for all values of s 9E so in the disk ls — sol < 6 (5 > 0). Note that s may approach so from any direction in the complex plane. EXERCISE Show that the limit of F (s) = s2 as s ——> 0 is zero. Note that the value of F (s) at s = O is immaterial. DEFINITION 3: The fimction F (s) is said to be continuous at s = so if F (so) is defined and lim F(s) = F(so) (C-8) s—>so EXERCISE Show that F (s) = s2, all s, is continuous at s = 0, and in fact for all s. DEFINITION 4: A function F(s) is said to be differentiable at a point s = so if the limit F + A — F F/(SO) = 331310 W (C_9) exists. Letting s = so + As, we may also write this as F s — F s F’(so) = lim ~—(—)———(-0—) (C-lO) 3—)50 S — S0 Note that s may approach so from any direction and the resulting limiting values of (C-10) must be the same regardless of the direction of approach. 568 App. C / Functions of a Complex Variable—Summary of Important Definitions and Theorems EXAMPLE C-2 Consider the derivative of the complex function F (s) = s2. We find that . s + As 2 — s2 A 2 = lim 2s 5 + (AS) As—>0 As = 25 (C-11) at any point s. At this point, we simply state that all the familiar rules for differentiation of functions of a real vari- able continue to hold for functions of a Complex variable. For example, the derivative of sin s is cos s, etc. The following example gives a function that does not have a derivative anywhere. EXAMPLE C-3 Consider F(s) = s* = 0' — ja). For this function, F(s + As) — F(s) _ (0' + AU) — j(w + Aw) — (0' — jw) As _ Ag + jAw _ Ao—jAw _ Arr + jAw (C-12) where As = A0- + jAw. The limit of this ratio depends on the way As approaches zero. For example, if Ao- —> 0 first and then Am —9 0, we obtain as the limit — 1. On the other hand, if Aw —> 0 first and then Ao—> 0, the limit is +1. Hence this function is not differentiable anyplace, since s was arbi— trarily chosen. DEFINITION 5: A fimction F (s) is said to be analytic (regular) at a point s = s0 if it is defined and has a derivative at every point in some neighborhood of so. Note that this definition is stronger than simply saying that a function is differentiable at a point, since the derivative must exist everywhere in a neighborhood of the point. DEFINITION 6: A function F(s) that is analytic at every point in a domain D is said to be analytic in D. (Note: A domain is simply an open, connected set of points.) DEFINITION 7: , A function F (s) that has at least one analytic point is called an analytic function. DEFINITION 8: A point S0 at which the analytic function F (s) is not analytic is called a singular point. It can be shown that an analytic function F (s) of a complex variable has derivatives of all orders, ex- cept at singular points, and can be represented as a Taylor series in a neighborhood of each point sO where F (s) is analytic: 1» (n) F(s) = 2—) F (3°) (5 — so)" (013) App. C / Functions of a Complex Variable—Summary of Important Definitions and Theorems 569 where F(")(so) denotes the nth derivative of F (s) evaluated at s = so. For example, (5 _ _____s__no)n =ie° (014) about any point so in the complex plane. In fact, this IS how the transcendental functions e3, sin 5, cos s, etc., of a complex variable are defined. The function F (s) = G(s)/(s — so)“, where G(s) is analytic at s = so, is singular at s = so because of the factor (s — so) ‘M. Such a singularity is called a pole of order m. This type of singularity is said to be isolated and, in the neighborhood of an isolated singularity, the representation 00 F(s) = E Cn(s — so)" (C-15) n=—OO can be used. More specifically: DEFINITION 9: If there is some neighborhood of a singular point so of a function F (s) throughout which F is analytic, except at the point itself then so is called an isolated singular point of F. The series (015) is known as the Laurent series expansion for the function at s = so. If C" = 0, n < —m, F(s) is said to have an mth-order pole at s = so (m > 0); if m = 1, F(s) is said to have a sim- ple pole at s = so; if C" ¢ 0 for infinitely many negative values of n, F (s) has an essential singularity at s = so. Examples of functions with isolated singular points are Us and eS/(s — 1)2. The function l/sin(7r/s) has isolated singular points at s = :1, t 1/2, t 1/3, . . . . At s = 0 this function has a non- isolated singular point since every neighborhood of s = 0 contains other singular points. THEOREM 1: A necessary condition that a function F(s) = U + jV of a complex variable s = 0' + jw be analytic at a point so is that its real and imaginary parts satisfy the Cauchy-Riemann equations 6U 6V 8U 6V 60' do; 6m 60' ( ) THEOREM 2: Sufiicient conditions for a function F (s) of a complex variable 5 to be diflerentiable at a points = so is that U(o-, w) and V(o-, (0) have continuous first partial derivatives that satisfy the Cauchy- Riemann equations at the point s = so. THEOREM 3 (CAUCHY’S INTEGRAL THEOREM): Let a function of a complex variable F (s) be analytic everywhere on and within a simple closed curve C (by simple, we mean that the curve does not cross itself). Then 93 F(s) ds = 0 (017) C where 515C (-) ds denotes the line integral in a counterclockwise direction along C. Figure 01 illustrates the circumstances of the theorem. Suppose that F (s) has an isolated singular- ity at s = so which is inside C as shown in Figure C-Z. Is fie F(s) ds = 0 still? No, because F(s) is not analytic everywhere within and on C. However, consider the simple closed curve shown in Figure C-3. Since F(s) is analytic within and on the simple closed curve C + C1 + C’ + C2, it follows by Cauchy’s integral theorem that 36 F(s) ds = 0 (C-18) C+CI+C'+C2 570 App. C / Functions of a Complex Variable—Summary of Important Definitions and Theorems iw , F (s) analytic FIGURE C-1. Illustration of Cauchy’s integral theorem. Breaking the line integral up into the sum of four integrals over the separate curves C, C1, C2, and C’, We obtain J F(s)ds + f F(s)ds + f F(s)ds + f F(s)ds = 0 (019) C C1 C2 0 . Now, as the gap between the curves C1 and C2 approaches zero, the second and third integrals will ap- proach equal magnitude, opposite in sign values (both have the same integrand, are integrated over the same length; but are taken in opposite directions). It follows, therefore, that i ()3 F(s) ds = —4) F(s) ds, 5 —> 0 (C-20) C . 'c' We now focus our attention on the integral over C’ as 6 ——> 0. For any point s on C’, we may write s = s0 + 8ei9 (s on C') (C-Zl) it.) {C F(s)ds ¢ 0 C ‘ \Fiic-Z. Line integral around a closed contour that is not zero because of a singularity within tour. App. C / Functions of a Complex Variable—Summary of important Definitions and Theorems 571 iw so denotes an isolated singularity 0 FIGURE C-3. Composite closed curve, C + C1 + C2 + C’, chosen so as to satisfy Cauchy's integral theorem. Therefore, in the integral over C ’ of (C-20) we may substitute ds =j6ef9 d0 (C-22) which results by taking the derivative of (C-21) (note that for any point on C’, both s0 and 8 are fixed). Also, we may express F (s) in the neighborhood of s0 as a Laurent series as given by (C—lS). In particu- lar, on the curve C’, s is given by (021), so that s — so = 5:319 (023) and (C-15) may be expressed as F(s) = i deem)" n=-m E Cn6"ej”” (s on C’) ((3-24) n=—w Substituting (C-22) and (C-24) into the integral over C’, we obtain 0 oo 98 F(s) ds = f [ 2 C”6”e’"9] Jae/'9 d0 (025) C’ 0: 27r n = —00 where the line integral can be written as an integral over the angle 0 because of the circular path. Note that the integral over 6 goes from 0 = 277 to 9 = 0 because of the clockwise orientation of C’. Because the Laurent series for F (s) is uniformly convergent to F (s) in the neighborhood of so, the integral and sum in (C-25) can be interchanged to yield 2n 93 F(s)ds = — 2 ansm f ef<"+1>0d0 (C-26) C, n=—w 0 where the integral is now taken from 0 = 0 to 0 = 277 with a minus sign inserted to account for the re- versal of limits. Now, for n + 1 ¢ 0, the integral of exp[j(n + 1)0] from 0 to 277 is zero (the student 572 App. C / Functions of a Complex Variable—Summary of Important Definitions and Theorems may show this by integration and substituting the limits). For n + 1 = 0, the integral in (C-26) evalu- ates to Zn. Thus all terms in the sum are zero except the term for n = -— 1, which gives Sims) ds = — 2717a} (027) Therefore, as 6 and e approach zero, (C—20) evaluates to SI) F(s) ds = 2717'C_1 (C-28) C where C_1 is the residue of F (s) at s = so. If F (s) has more than one isolated singular point within C, say at the points s1, s2, . . . ,sN with residues K1, K2, . . . , KN, then we may follow the procedure outlined above for each singular point and arrive at the residue theorem: THEOREM 4 (Residue Theorem): Let F (s) be analytic everywhere within and on the simple closed curve C except at the isolated singular points s1, s2, . . . , sN. Then N 96 F(s) ds = 2777' 2 K, (029) C n=1 where fic (-) ds is the line integral in a counterclockwise sense of F (s) around C and K1, K2, . . . , KN are the residues of F (s) at s2, s2, . . . , SN. We complete this appendix with several theorems pertaining to the single-sided Laplace transform. THEOREM 5 (Absolute Convergence of the Single-Sided Laplace Transform): The integral X(s) = f: x(t)e_" dt (c—30) converges absolutely for all Re(s) > c if LL |x(t)| dt 5 K < oo (C-31) for0<L<°°and0<K<00, and he) i s Ae“, t > L (032) for some real A and c. Proof. According to the definition on absolute convergence, we must show that I = J Ix(t)e‘”l dt < 00 for o- = Re(s) > c (C-33) 0 We divide I up into two integrals as L 00 I = f |x(t)|e“"dt + f |x(t)|e‘”'dt 0 L = 11 + 12 (C-34) Now L 115 eML f ix(t)| dt 5 Kel"'L (035) 0 “WWW App. C / Functions of a Complex Variable—Summary of Important Definitions and Theorems 573 and co Ae—(a—c)L 12 s A I e_("_c)’ dt = ~— since 0' > c (C-36) L a- — c Hence, Ae—(a—c)L I=Il+Izs e|e|L+————<oo fora->c (037) 0' — C DEFINITION 10: (Uniform convergence of an integral): The integral I(s) = f: g(t, s) dt (C-38) is said to converge uniformly for s in a specific range (say 0 to 00) if for each 6 > 0, we can find a value d such that < e for b > d (C-39) [b g(t, s) dt — I(s) THEOREM 6: If the integral X(s) = [wane-5' dt (040) 0 converges absolutely for s0 = 0'0 + jwo, it also converges absolutely and uniformly in the half plane Re(s) 2 0'0. THEOREM 7: If the single-sided Laplace transform integral converges absolutely for o- > 0'“, then X(s) is analytic for 0' > O},- This theorem tells us that the singularities of X(s) must lie to the left or on the line Re(s) = on. By a process known as analytic continuation, X(s) may be defined and shown to be analytic on the entire s-plane except, of course, at its singularities. THEOREM 8: If X} (t) and x2(t) both have the same single-sided Laplace transform, then x1(t) = x2(t) (C-41) except, possibly, at a set of points with “measure zero”; that is, the integral of Ix1(t) — x2(t)l over the range of integration is zero. Further Reading The books by Churchill and Kreyzig cited as references in Chapter 5 are recommended if the student wishes to have a more complete treatment of complex variable theory than that given here. ...
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