ECE3337NOTES3_20081015125700

ECE3337NOTES3_20081015125700 - 4.5 an r we“ am" we'J...

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Unformatted text preview: 4.5 an r . we“. am" we ' .'J 4 r, \.......»‘..... :ifihhgaanimm. n1; Tina m‘ ' w? some.” : ii , _l- man-mitt; Jul-2.15.5“ . {xi-.u;.:ufism..- (~34ng -.-ggl,-,,,.,_‘;u=~ “iv-aim» 1,"... Chapter 2 VARIABLES AND FUNCTIONS A symbol, such-as z,'which can stand for any one of a set of complex numbers is - called a samples: variable. . If to each value '_ which a Complex variable 2 can assume there corresponds one or ' .mOre values of a complex variable 10, we say that w is a function of _z and write w =. f‘(z) ' Z'or- 'w: 6(2), etc; The variable 2 is sometimes-called an independent amiable, while w is called a dependent Wfidble- The value of a fflflCtion' at .2 2a. is often written flu). Thus- if N?) 7—: 22, then 'f(2£) = (23')2 2 "4, - ' . 4 ._ .. I SINGLE AND 'MULTIPLE-VALUED FUNCTIONS If only one value of 3w corresponds to each value of 2, We say' that w'is a singlewulned ' -‘ function of z or that flz) is single~v‘alued. If/more than one value of 7w corresponds to each value of 2, we saygtliat wiisj'a multiple-valued or manyfualned, function of z. _ ' ,‘A'multiple—valued function can be considered as a collectionlof single-valued functiOns', , I each member of which is called a branch of? the function. 'It is customary to'consider. one ' ' particularfmember'as-axprinc'ipal branch of themulti'pleavalued function and the value of " - theiunction ”corresponding-i 'toj-this' branch as th'e‘pninc'ipql’nblne; ' \ Examine 1; _ If to = 2”, than to each value of 2 there isonly uneavalue‘ of w. Hence 150F112) = 23‘ is a - ' x : ' single-valuedifiunction of z. ‘ ' , " ' ‘ ‘ A __ ‘ a-multipledvalued (in-this case two-valued) tunction'ofla. '- , _ _ _ _ . ' Whenever we speak of function we shall], unless otheerise stated, assume single-rimmed ‘ Egan-113192: - -A"I:II toga/2, theniito‘each value of z thererare Novelties of 10} 'Hence w: f(z) =-_'z'1/5_3‘is INVERSE'FUNCTNNS' ,- * _. - s ' n~ 1 ‘ - ' 1 ‘ , If n) = f(z), then we can aISQ consider z as a function of w, written 2 = g(w) = f" (w). The function f“ is'roften called the inverse function corresponding to f._ Thus w = f(z) and w =‘_f‘1(z)r_are inverse functions of each other. ‘ TRANSFéRMATIONS - If w :15 +131; (where stand a are real) is a singleevalued function of z ':o:r+iy (where ' I ' sand 3; are real),rwercan write n+2’v ’2 f{_x+iy). Byequatingreal and imaginaryparts ' this is seen to beequiv’alent to Thus given a point (any) in then' plane, such as P in Fig, 2-1 below, there corresponds a point (n, 1'2) ‘in the to plane, say P’ in Fig.. 2—2 below.."The 'set of equations ,(1) [or the , equivalent, w : f(z)} is called-a transformation. 'We Say that point P is mapped or ~ ' transfonned into point P’V'bygmeansof the transformation and call .P’ the imageof ‘P. 33‘ 34'. ' . FUNCTIONS, LIMITS AND CONTINUI‘I‘Y {crime 2 Example: If to = 22, then u + 2'1: 2 (2: +139)?- 2 22 — y: + 259:?) and. the transfofination is u : 22—y2, o : 23y. The image‘ of a point (1,2) in the 2 plane is the point (—3,4),in the in plane. ' ' Fig- 2-1 . ' _ Fig. 2-2 Ingeneral, under a transformation, a set of points such asthose on curve P6) of Fig. 2-1 is mapped into a correspbnding set of points, called the image, such as those on curve P’Q’ in Fig. 2—2. The particular'characteristics cf the image depend of course on the type of function re), which is sometimes called a mapping function. If/f(z.) is multiple- valued, a point (er'curve) in the 2 plane is mapped in generalinto more than one- point (or curve) in the or; plane. ' CURVILINEAR COORDINATES Giventhe transformation w = Hz.) or, equivalently, u 2 u(x,y), o i okay), we call (x, y) the rectangular coordinates corresponding to a point P in the 2 plane and (u, o) the , curvilinear coordinates of P. 2 plane ' w plane Fig.2-3 ' t 1 Fig. 27-4 The curves u(a:,y) = 0;, 142:, y) 2 c2, where c: and 62 are constants, are called coordinate curves [see Fig. 2-3] and each pair of these curves intersects in a point. These curves map into mutually orthogonal lines in the ‘w plane {see Fig. 2-4]; ' THE ELEMENTARY FUNCTIONS 1. Polynomial Functions are defined by .w : aoz" + (112“"1 + ' ii + (tn—12 + an = 13(3) (2) Where do aé 0, a1, . , .1, on are complex constants and n- is a positive integer called the degree of the polynomial P(z). ' ' The transformation on = az +1) is called a linear transformation. J i CHAP. 2] - r _FUNCTIONS, LIMITS AND CONT-INUI-TY 357 2. Rational Algebraic Functions are rdefinedby, ' P_(z) ‘ , a . - - 2’ 0(2) ( 1 WhereIP(z) and 9(2) are polynomials. We sometimes call (3) a rational transfermation 112 + 1) oz + d .tional linear transfdrmation. , The special case w I‘— where ad— be aé 0 is often called a bilinear- or frac- 73. EXponential Functions are defined by w — —ez : 32“” 2 eicosy—Fisiny) I I (4) where e— — 2. 7I1828.. . is the natural base of logarithms. If a is realiand- positive, we define 7. _ 1. - 1 . . I a: ._.. 8: na . . (5) ' Where In a is the natural logarithm of a. This reduces to (11) if a— — 9. Complex exponential functions have properties similar to those of real exponential- functions. For example, 62: - 322—- — ezr+2=, 321/32?— — 321-22_ - ‘4. Trigonometric Functions. We define the trigonometric or circular functions sinz, cosz, etc” in terms of exponential functions as follows. sinz— (HM—e ‘ cosz—eiz-Fe—i'z , ‘ 2i 1 _ 2.. 1 ' - secz— 1 - 2 cscz—i— 2% _‘ cosz _ 32+?“ ' . sin-z 9273'?“ - sinz 'eiz—e—‘2 cos'z , “2+?“ stanz;-'——=sz—'—2; cotz= 3 =L;—_£T) ‘- - cosz 2(3 +3“ )' , slnz ‘ e ——e-‘ Many of the properties familiar in the case of real trigonOmetricI functions also _ hold for the complex trigonometric functions. For example, we have ‘ sin2z + cos2z = 1 1 +tan2z 2 'sec2z - 1 + cot2z— — csc2z sin (-2): — sin 2 ' cos (#2) =' cos 2: tan (—z)‘= ~— tan '3 ’ " sin (2;: 22) = sin 21 cos I22 + cos 2; sin 22 t ' cos (2; if 22) m I _cos 21 _cos 22 '1— sin 2; sin 22 "tan21i'tan22 ' ' 7 , tan (21 1- 22); =, —-—’-——— 1 I tan 21 tan 22 ' 5; Hyperbolic Functions are defined as followsr . e2 —_e‘2 - . e“ + 6'2 smh Z = , -~-—---— ' 1 cosh z : ___... J ' . = '. 2 _i I . 1 2 I ‘ . _ 1 -_ 2 _ '- , __ 1 _ _ 2' secha ___ coshz _ W cschz _- sinhz —' 322—972 tanh z- _ sinhz _ 9’ ‘73" (:0ch _ coshz ___ 62+?“ coshz e2 + e"; I ' sinhz _e.‘ —‘ 3'3" ' The following properties hoitLI cosh2z —' sinh2z z_-= 1 1 —— tanh2z # se‘chl2z‘ coth2z — 1 =7csch2z sinh (—2): —' sinh z cosh (—z) — cosh z. tanh (~12): ,._- tanh z sinh (211- 22) = sinh z: cosh 22+ _- Cosh 'z; sinh— 22 ' 'I cosh (z; i. 22) - = cOsh zi cosh 22 i sinh a; Vainh 2:2 tanh (21 i 22) = -, tanh Z] i tanh 22 V . 1 it tanh z: tanh 2'2 It 36 FUNCTIONS, LIMITS AND CONTIINUITY [0111113. 2 The following relations exist between the trigonometric or circular functions and the hyperbolic functions: sin iz .2 i sinh 2 cos iz = cosh zr tan is i tank 2 sinh is = i sin; cosh it cos 2‘ I tanh is _= i tan 2 ll 6. Logarithmic Functions. If 2 = c”, then we write in 7- ln‘a, Icalled the natural toga,- rithm of 2. Thus the natural logarithmic function is the inverse of the exponential function and can be defined by _ I w 2 Inc» = lnr+i(0+2kw) kEO,+1, +2,. where z— —- rema — WWW“), Note that 1112 is a multiple-valued (in this case infinitely- . many-valued) function The principal-value or principal branch of lnz is sometimes defined. as lnr + it) where 0_ S 6 < 211'. However, any- other interval of length 271' ‘can be used, e. g. —nI' < 6_< ... 71', etc. - I The logarithmic function can be defined for real bases other than e. Thus if 2:51“, then wzlognz where c>0 and 11,950, 1. In this case z=e"’hm and so 10- —— (1n z)/(ln 0,). ‘ , 7. Inverise Trigonometric Functions. If z— -- sinw, then 10— — sin 1,2 is called the inverse sine of z or are sine cf 2. Similarly we define other inverse trigonometric or circular functions eds 1z,tan"12:, etc. These functions, which are. multiple-valued, can be expressed in terms of natural logarithms as follows. In all cases we omit an additive I constant kai, k: 0, +1, +2,. ., in the logarithm. sin'lz = —.Iln (iz+1/1-— 22) Icsc"1z “If”: ilanl/zE—l) see“‘z lln(1+‘f:?) 3 1 1+iz I 1 2+2? 1 —. 7 —]_ . tan z —2i in (12.2) out a —.2?’ 111(z _ i) 8. Inverse HYPerbolic Functions. If z :sinhw then 20 = sinh—lz is called the inverse I hyperbolic sine of 2. Similarly we define other inverse hyperbolic functions cosh 12;, tanh 12, etc. These functions, which are multiple-valued, can be expressed in terms of natural logarithms as follows. In all cases we omit an additive constant 219.5, k: 0, +1, +2,. . ., in the logarithm. I sinh‘I z = 1n (2 + 1/zz+ 1) csch’l 2 In (L ”22“) cosh"2 = 1n(Z+\/2"-1) seth—lz = m(_1+ 1—22) 2 _ 1 1+2 , I 1 2+1 1 —— .. — _ _ tanh z —,21n(1_z) coth lz __ 2]n(z_1) 9: The Function 2", where a: may be complex, is defined as 8““ 2. Similarly if f(z) and 9(a) are two given functions of a, we can define 112)”) = eg‘z’inf‘z’. In general such func- tions are multiplevalued. 10. Algebraic and Transcendental Functions. If it: is a solution of the polynomial equation Po (2)112" + P; (2)21)“1 + - - ' + Pu—1(z)w + Fri (2) = OI (6) where Po $ 0, 131(2), .. ,P (z) are polynomials in z. and n is a positive integer, then in: f(z) is called an algebraic function of 2. Example: 111*“ —- 2W is a solution of the equation wz—z = G and so is an algebraic function of z. ' .33 . V FUNCTIONS, LIMITS (AND CONTINUITY ' ' - {CHAR 2 is the limit of flat} as 2 approaches Zn and write lim f(z): _ if for any positive- num- ' 2-DZ“ ber e (however small) we can find some positive number 8 (usually depending on e) such that |f(z)—l| < 6 whenever 0 < |2—Zo[ < 8. In such case we also say that f(z) approaches 1 as 2' approaches an and Write f(z) ->l as a '-> 20. The limit must be independent of the manner in which it approaches 20. Geometrically, if 20 is a point in the complex plane, then lim f(z) it if the difference 2—92“ in absolute value between f(z) and t can be made as small as we wish by choosing points z sufl‘iciently close to 20 (excluding z: .20 itself). ' - ' 2 5e ' . . . Example: , Let Hz) =. {3 :n :3. Then as 2 gets closer to 1' (i.e_.-a approaches it), f(z) gets closer ' to £2: —1. We thus suspect that lim flz) = —-1. To prove this we must see whether z-bi the above definition of limit is satisfied. For this proof see Problem 23. . , Note that lim f(z) at fit), i e. the 'Iimit of {(2) as 2 -* i is not the same as the value z_-+i of flz) at zi—“i, since f(t')= 0 by definition. The limit would in fact be -1 even if f(z) were not defined at z=i. , 7 _ '1 When the limit of a functionexists it is unique, i.'e. it is the only one.(see Problem 26); If f(z) is (multiple-valued, the limit as 2 +20 may depend on the particular branch. _ THEOREMS 0N LIMITS 7 I If. lim f(z) z A and lim 9(2) = B, then zo-bzo , z-+zo 1-, 1im{f(z.)+9(2)} = lim f(2) + lim 9(2) =‘ A +'B — . 2"30 z-an erzo , 72- 351: {f(Z)—9(Z)} = hm f(z) — lim 9(2) = A _ B ' 3_. 133° {sages = {5333(2)}{132 9(2)} = A3 lim f('z) - f(z) .: zuz. _ A . _ 4-lilgng(z)- 3129M.— E 1: BTU . INFINITY . By means of the transformation 10— — 1/2 the point 220 (i. e. the origin) is mapped into ur- — 00, called the point at infinity in the to plane. Similarly we denote by z— — no the. pointet infinity in the 2 plane. To consider the beha’vior'of flat). at z = no, it suffices to let ' z: ' 1/w_ and examine the behavior of f(1/w) at 1020. ' We say that lim f(z)— —— l or f(z) approaches 1 as z approaches infinity, if for any 2—»: e) 0 we can find M > 0 such that [f(z) — ll < 2 whenever |z] > M. We say that lim f(z)— on or f(z) approaches infinity as a approaches 29. if for any Z-Dzo 'N>0 we can find 8>0 such that |f(z)( > N whenever 0 < [z— 20} < 8. ' CONTINUITY Let f(z) be defined and single-valued in a neighborhood of 2220 as well as at 2:20- (Le. in a 8 neighborhood of 20). The function f(z) is said to be continuous at z=zo if Iim f(z) : f(Zo)_. Note that this implies three conditions which must be met in order that 2H2“ _ " ‘ f(z) be continuous at z: 20: ' h'm'iserwt'qlfi'ira A slow-aw“ 'a-ai‘fz- 1'7? altim- "mien: .. -:‘ '- ‘ point a.) such that |f(z ‘7 on e-lbutnot onthe particular point ao,-We say that flz) is uniformly continuous inthe region. CHAP.‘ 2] ‘ _' FUNCTIONS. LIMITS AND CONTINUITY ' . 39 -1.' lim f(z} = l must-exist '2. flag) must exist, i.e. 7(a) is defined at at 3.. l=f(Zo)- ' .- v ' I Equivalentlfiif flz) is- continuous at 2.} we can write this in the suggestive form lim f(z)i = f(lim z) . ‘ , . ‘ . z-ozo z—rzo ‘ 2 a; - .7 - , - Example. 1: If f(z) = S :=; hen from the Example on Page 33, gig: re) = —1_. But. no = 0. Hence lim fix) 9* f6 and the function is not continuous at z: i. - , z-+£ _ l . Example 2: ' If f(z) =z2 for all 2, then ling fl?) =7 fl!) = —'1 and fix) is continuous at 2:11 z—r _ - . . Points in the z plane where fizj' fails to be continuous are called discontinuities of fizz); and f(z) is said to be discontinuous at'these points. If iim flz) exists but is no equal to f(z.)), we call '20 anemooable discontinuity since by redefining. f(zoi.to be the sanie as -lim7f(z)r thefunction becomes continuous. ' z-srzo Alternative tothe above definition of continuity, we can define f(z) as continuous at .- 23:20- if for any £>0 we can find 8>O such that 1f(z)— f(zo)| < 6 Whenever [2—29] <8. ' Note that this is simply the definition of limit with l=-f(zo) and removal of the restriction thatZSéZo. ‘ I I 'I ' ' 5 7 To exainine the continuity of 112) at z = on, we place 2: llw and examine the continuity ' .of fillw) at w=0. ' . - CONTiNUITY IN A REGION .- - A functifm flz) is said to be continuous inn region if it is continuous at all points of the regiOn. . . _ _ _ _ THEOREMS on CONTIfiUITY _ i 7 u , _ . _ ‘- ' Theorem L If f(z) and- 9(2). are continuous at 23:20, so also are the functions '.,lf‘(z)'+ 9(2)? fig) -901), 1139(2) and. iii—i, the last only if.g(zo)_ a? 0_. VuSimilar results hold for "iqsiinuity in a resibm 7 ' ., ' ' f I 9' I '- Theorem 2; Aniongthe functions centinuous in every 'fin'ite region are (a) all poly- _ 'nomials, ”(b) 32, --(c) sin 2 and-cos z,‘ i " Theorem 3. If- w¥ f(z) is continuous at 2:20 and dz=g{§) ‘is continuous at 22;]; and-ital": 9030); thenjthe function w 5: {[96]}, called a function of a function or, composite function, is continuous at C :41}. This is simetimes briefly stated as: ' A continuous function - of a continuous function is continuous. ' . Theorem 4. If f(z) is continuousin a closed region, it is bounded in the region; i.e. _ there‘exists a constant M such that |f(z)] < M for all points z of the regidn. ', ,- “Theoremx5. If flz) is Continuous in a region, then the real and'inlag'inary parts of fin} arealso continuOus in the, region. - ‘ " UNIFORM CONTINUITY _ ; . _ : ' ' Let Hz)" be continuous in a region. Then by definition at each point Zn. of the region andforany'e > 0, we can find 3' > 0 (which will in general depend on both 6 and’the particular _ ):— f(zo)[ < {whenever [szd < 8. If We can find 8 depending ...
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