ECE3337NOTES1_20081015125534

ECE3337NOTES1_20081015125534 - WPMES Mug 4...

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Unformatted text preview: WPMES Mug _. 4 _ _. COMPLEXNUMBERS , '- [cane-1 SOmetimes we refer to the a: and y axes as the real and imaginary axes respectively and to the complex plane as the a plane. The distance between two points 251.— —- x1+iy1. and 22— — xz-H'yz in the complex plane is given by |z1— 2217— — 1/(x1—mz)2+ (yr—1M2. POLAR FORM OF COMPLEX NUMBERS If P is a point in the complex plane corre- ‘ sponding to the complex number (any) or x + iy, then we see from ”Eigfil—P, that .'_. aI' _ ‘ x— recs/y— rsmo where r— — (MM—'11P: |x+iyl is called the modulus or absolute value of z— — :1: +121; [de- ,. noted by mod 2 or _|z|];. and 9- called the ampli- tude or argument of 2—- - x+iy [denoted by , erg z], is the angle which line OP makes with the positive x axis. It follows that z m —x+iy_— '- r(cos& +1s1n3) (1) which is called the polar. form of the complex number, and r and 0 are called polar coordi— notes. It" is semetimes convenient to write the Fig' 1'3 abbreviation cis 0 for cos 6 + 1' sin 6. ‘ Fer any complex number Zréo there Corresponds only one, value of ' 9 in 056(211. However, any other interval of length 211. for example —1r<6‘En-, can be used. Any' particular choice, decided upon in advance, 1s called the principal range, and the value of 9 is called its principal value. ‘ DE MOIVRE-‘S THEOREM If 21- —* x1 +iy1— — r1 (cos 91+ isin 91) and 22— — x2+iy2— -—- rz (cos 92 +isin 02), we can show that [see Problem 19] 2122 2‘ n+2 {cos (61 -i_- 92) + i sin (01 + 92)} ' (2), z r . . . - ‘ E; 2 1.7; {cos (01 - 62) + z 8111(61 — 62)} I . ' ' (3) 'A generalization of (2) leads to . , I 21226-2,P = r1r2---r,. {cos(l91+92+ -- - +91.) + isin(01+62+ ..,.+ 6.1)} I (1;) and if z1=z2,= :zuzz this becomes 2 l ' I i z"— — [r(cos 9 + i sin 6))" which 13 often called De Moivre’ s theorem. 1-" (cos 119 + 1' sin n0) ' . ' (5)" ROOTS or COMPLEX NUMBERS ' ' A number 10 is called an nth root of a complex number 2 if w“=z, and we write ‘ / 10:21“. From De Moivrefs theorem we can show that if n is a positive integer, zllfi‘f': {r(co_s'6 +13s1n9)}“" ' ' ' _ “ I = r“’-‘-{c cso 3(9 +fk7r) +_ win (9 fight» ion: 0, 1, 2, . . ..n—1- - (5) from which it follows that there are at different values for z“", i e. 11 different nth roots of 2, provided zeéo. : l -_-...,,, fl _, ._ , ,,J CRAP. I] I ' COMPLEX NUMBERS 5 ' EULER’S' FORMULA ' L By assuming that the infinite‘series expansion eI = 1+x4; 32/2!'¥I—a3/3!+ - -- of , elementary calculus holds when x=i6, we can arrive at the result ' ' 7%, em = cosd +isin9 9:2.71828...' ' (7) which is called Eulea’s formula. It is more convenient, however, simply to take (7) as a definition of e“. In general, we define ‘ k % 65"": ex“? = e‘e‘” = lax-(cosy + i sin y) - ' (8)- -'In the special case where il=0 this reduces to (2‘. Note that" in terms of (7;) De Moivre’s theorem essentially reduces to (#9)“ = cw; POLYNOMIAL EQUATIONS - ' Often in practice werequire solutibns of polynomial equations having the form _‘ ate“ + men—1 + age"2 + --- + (Ia—12 + a1' = 0 '(9) where aoeéO, a1, '. . ;,_a,. ' are given complex numbers and n is a positive integer called the degree of the equation. Such solutions are also called zeros of the polynomial on the ' left of (.9) or roots of the equation. A very important theorem called the fundamental theorem of algebra [to be proved in Chapter 5] states that every polynomial equation of the form (9) has at least one root Vin-the complex plane. From this we can show that it has in fact a complex roots, some or all. of which may be identical._ ' If 21,32, . . -,Zn are the a roots, (.9) can be written _ . ' do(zi.‘21)(ze22)'v--(z—z») = 0 ' 7 j , , (10) which is called the factored form of the polynomial equation. Conversely if We can write - (9) in the form (10), we can easily determine the roots. _ . ' THE nth ROOTS 0F UNITY. , / _ The Solutions of thef'equation 'z"=_1 where n is a positive integer are called the nth roots of unity and are given by; _ _ L ' ‘ _ ' a =_ COSZkrrl‘n +‘isin2kw/n '; 92w" 1.: .—. 0,1, 2, ..'.,n—1 '(11) If we let a) = cos-2«/n—+isin 21r/1‘t gem», the a roots are 1, a), :92, ...,m““. Geo- . metrically they represent the n vertices of aregular polygon-of 1.: sides inscribed in a ciréle of radius one with center at the origin.‘ This circle has the equation [2]: 1 and is often called the unit circle. , - VECTOR INTERPRETATION ‘OF COMPLEX NUMBERS A complex number a m_ a: +iy can be con- y _ _ sidered as a vector OP Whose initial point is the - 4 ‘ 7. B origin 0 and whose terminal point P is the / ‘ point (any) as in Fig.'1-4. We sometimes call A 7 ‘ - 0P = a: + iy the position vector of P. Two vec- ' tors having the same length or magnitude and direction but different initial points,asuch as DP and AB in- Fig. 1-4, are considered equal. . Hence we write 01’ = AB = a; +iy. 7 7 A _.Fig.1-4 CHAR 1} \— COMPLEX NUMBERS _ , . 1 7 COMPLEX CONJUGATE COORDINATES A pOint in the complex plane can be located by rectangular coordinates (x,y) or polar coordinates (1,6)1Many other possibilities exist. One such possibility uses the fact that .11- .— §(z+z), 312%(1—91) where z— - x+ig. The coordinates (2,2) which locate a point are called complex conjugate coordinates or briefly conjugate coordinates of the point [see Problems 43 and 44]. , POINT SETS Any collection of points in the complex plane is coiled a (two-dimensional) point set, and each point is called a member or element of the set. The following fundamental definitions are given here for reference. 1. Neighborhoods. , A delta, or 3.111191110110011 of a point 211 is the set of all points z 3. '5. 6. 10. 11. such that [z— 211] < 8 where 8 is any given positive number. A deleted '8 neigh- borhood of so is a neighborhood of an in which the point 211 is omitted, i e. O < E2"Zol < 8. ' Limit Points. A pOint so is called a limit point, cluster point, or point of accumu- lotion of a point set S if every deleted 8 neighborhood of 20 contains points of S. Since 8 can be any positive number, it follows that S must have infinitely many points. Note that 20 may or may not belong to the set S. Closed Sets. A set Sis said to be closed if every limit point of S belongs to S, i. e. if S contains all its limit points. For example, the set of all points 2 such _ that ]z|_ S 1 is a closed set. Bounded Sets. A set S 1s called bounded if we can find a constant M such that |z[ < M for every point z in S. An unbounded set is one which is not bounded. ' 77 A set which is both bounded and closed is sometimes called compact. Interior, Exterior and Boundary Points. A point 2., is called an interior point - of a set S if we can find a 8 neighborhood of 20 all of whose points belong to S. If every 8 neighborhood of 20 contains points belonging to S and also points not. ' . belonging to S, then an is called a boundary point. If a point is not an interior or boundary point of a set S, it 1s an exterior paint of S. Open Sets. An open set is a set which consists only of interior points. For example, the set at points a such that [z] < 1- is an open set. ‘ Connected Sets. An open set S is said to be connected if any two points of the set can be joined by a path consisting of straight line segments (i. e. a polygonal path) all points of which are in S. Open Regions or Domains. An open connected set .is called an open region or f - \ domain. Closure of a Set. If to a set S we add all the limit points of S, the new set is _ called the closure of S and 1s a closed set. Closed Regions. The closure of .an open region or domain is called a closed region. Regions. If to an open region or domain we add some, all or none of its limit ' points, we obtain a set called a region. If all the limit points are added, the region is closed; if none are added, the region is open.111 this book whenever we use the word region without qualifying it, we shall mean open region or domain. _ ‘ COMPLEX NUMBERS 7‘ 12. Union and intersection of Sets. A set consisting of all points belonging'to set’Sr or set 32 or. to both sets S; and $2 is called the union of. S1 and Se and is denoted. by 31-1-43: 01' Si;.USe A set consisting of all points belonging to both sets 31 and S: is called- the ,- intersection of S1 and S: and is denoted by S182 or S1 n 82. 13. ' Complement of a Set. A set consisting of all points which do not belong to S is called the complement of S and is denoted by S. 14. Null Sets and Subsets. It is convenient to consider a set consisting of no-points-at ' 3.11.. This set is called the null set and is denoted by Q. If two sets S1 and Sa' . ' have no points in common (in which case they are called disjoint or mutually - esclusioe sets), we can indicate this by writing 31 n S2 _= 91:). Any set formed by chooSing some, all or none of the points of a. set. S is called. a subset of S. If we exclude the case ‘where all points of S are chosen, _ the set is called a proper subset of S. J 15. - Countability of a Set. If the members or elements of a set can be placed into a' one to one correspondence with the natural numbers 1, 2, 3,. ,the set is Called countable or denumemble; otherwise it is non-countable or non-denumemble. The following are tv'vo important theorems on point sets. 1. Weierstrass- Bolzano Theorem. Every bounded infinite set has at leaSt one limit point. ‘ - 2 Heine-Bore] Theorem. Let S be a. compact set each point of which is contained in one for more of the open sets _A1,A2, . {which are then said to cover S]. Then-- there exists a finite number of the sets 141,142, . . . which will cover S. Solved Problems FUNDAMENTAL OPERATIONS WITH COMPLEX NUMBERS 1. iPerform each of the indicated operations. (at) (3+2i)+(-7-i) = 3_-7+2i—~i = -—4+i- (l1) {HT—i) + (3+2i) = -—7 +3 —i+2i = ~4+ i The results (a) andlb) illustrate the commutative law of addition. (a) '{s- 60 (21—71: 8 ei-—21'+7 .= 15—8i (d) (5+31')+{(—1+2i)+(7-— 51)}- — (5+3z)+{——1+2i+7——5i}= (5+3i)+(6—— 31'): (e) {(5+3i) + (—1+2i)} + (7— 5i) = {5+3i—1-l-2i} + (7- 5i)— —' (4+5i) -l- ('7 5i): ‘ The results (d) and (e) illustrate the associative law of addition. 14—8i 14—85 I ('f) (2—3fl(4+2i) = 2(4+2i)—3i(4+2i) 8-i-4i-—12i—6i‘a 8+4i—12i—l-6 (g) (4+2i)(2-—3i);= 4(2—31‘) +2i(2-—3i) 8—12i+4i+6i2 8 —12i+4i-l?6 The results 0') and (9) illustrate the cmmututiue law of multiplication. (h) (2 — o{(—3 + 22105 — 41)} (2 o{—15 +125 + 101 — 812} 7 - (2—in—7 +222)_ — ~14 + 44s + 712 — 2212 = s + 511 (13 {(2— .i)(_3+2i)}(5“ 41') = {—6+4i+3i— 2i2}(5— 413) = (—4 + 7i)(5 - 4i) =7 —-20 + 1613 +3512 — 281'.2 = 3 -l- 5111 ‘ The results (It) and {i} illustrate the associative law of multiplication. [CHAR- ‘1‘ ' I L ‘fll—mf 1-4 _ 1 omanx NUMBERS ' , [oHA'PQI ' From the definitions of sum and :product o'n Page 3, we have - - ' (a,'_b) -=" (mo) + (0, b) = a.(1,0_) '+ 140,1). where . (mixer). = (0-0—1-1, o-1_+ 1-0) = (—11.0) By identifying (1,0) with ‘1 and (o, 1) with i, we see that (a, b) =. a+ bi, 15. If 21 :7(a1,b1), ,22 : (0,2,!)2) and za 7: (63,53); prove the distributive law: 21(22+23}.=7 2122 + 2123. ' ' ‘ ‘,We have 21(22 ‘1' 23) ll I ('11. bl){(a'2: 52) + 0713: 53)} = (a1, 51}{¢i2+ “a 152+ 53) {“1012 + a3) — 15192 + be), ddbz + be.) + 01(02 + ‘13)} ' (“1‘12 " 5152 + an“: — bibs: “"le + 51% + Giba + bins) ' (“was _ 5152- @152 + lhue» .+ (“1% " bibs: “lb-3+ blwé) , (“1: 511012: be) + (a1, biKGsi'bs) 1: 2122 + z133 [I H 'II POLAR FORM on COMPLEX NUMBERS 16. Eitpress each of'th‘efollt'ming' complex numbers in. polar form. (a)2+2\/§£ . \ ' ' ' ' Modulus-or absolute value, 1' = I2+2J§i1 =: m = 4. _ Amplitude or argument, ‘9 = sin“.1 2&4 = sin"1 xii/2:500 = vl3 (radians). Then _ 2+2\/§i ll r(cos 0 =1— i sin a) .=- 4(cos 60°‘+'i sin 60°) '- Fig-.‘I-Zz ’2 4(cos 7/3 + i sin 7/3) The result can also be written as 4 cis 1/3 or, using Euier’s formula, as 46"“. r =1—5+5£1= MA = 5J5 a. = 180° —- 45° 2 135° = 312-14 (radians) Then —5 +' 512 = 5V5 (cos 135° + isin 1359) ' ' = 5(2— cigar/4 = 5J5 chum (b). —-5 + 5% 1. ' . + = I-«é—m-i = «m = m o =isqo. + 30° = 210°. _= 7:16 (radians) 'Then- Avg—{2‘s = 2V5 (cos210° +isin210°) = 2V5 cis Me = zfiemfe («:0 ~31: A _ . - r = i—sq Io—ail = «0+9 = 3 a = 270° a 37/2 (radians) 3(cos-3r/2 + i sin Ber/2) _ = 3cis31r/2" = 333m '- ' Fig.1»25_ Then - —3i ' 1?. Graph each of the following: (a) 6(cos 240° + isin 240°), (11) 4331“”, (c) Zea—“m. ‘ {a} 6(cos 2403+ i sin 240°) 3' 6 cis 240° ; 6 cis 47/3 = 6 84"” can be represented graphically by GP in Fig. 1-26 below. I - _ . ' If we start with vector 011, whose magnitude is 6 and whose direction is that of the positive x axis, we can obtain OP by rotating GA counterclockwise through an angle of 240°. In general, re” is equivalent to a vector obtained by rotating 3 vector of magnitude 1- and direction that—of the positive 9: axis, counterclockwise through'an angle 9.' -’ Am stanzmitgzmma-ami w nr‘n—‘L'L‘ "win 41.134; - 4'- 24. Given a complex number (vector) 2, interpret COMPLEX NUMBERS 23. Prove the identities (a) sin3 6 = 4} sinfl — 1 sin 30, (b) cos.4 9: §cos49 +§cos26+%. I ' ' ' ' , . is ~59 ('41.) Adding (I) and (2), 958 47 3"“? = '2 case -. or cos 0 87 i; - - ' ' _ '9- #19 (b) Subtracting (2) from (1), eta 3—19 2,- sin a or s. a e1 2: . _ 3 __. (4115:1139 :7 (fig—fl) = M = ——{(ew)3 — were *9) + scene-“)2 461-998} ) n 813 _Si 4 2i 4 fi‘ sin 30 M Ma m .5 ‘3 | (b) cos.“ 9 ll (6‘9 + 9’”)! _ (as + rm)“ 2 _ 15 ' 1 . 1 _(e3‘9 — 38‘” + 36—” — e- aifl) = E (9‘9 '" 3—”) _ __1(e3|9 -- 9‘3“ 2'5 _ 1 - . 1" é4i°+074i9 - 1 o - = _. so 219 —2ifl —m = __ .-____, .. 16(e4_+4e_-l-6+4e +e ) 8( 2 ),+_2 __ l ' l E- -— 8c0340+2cosz2fl-[P-8 geometrically 28*“ Where a is real. Let 2:412“? be represented graphically by vector 0A in Fig. 1—30. Then zeia' ..—.. rem-gia_ — Mimi-c) is the vector represented by OB. Hence multiplication of a vector 2 by é“ amounts to rotating z counterclockwise through angle a. We can consider 6‘“ as an operator which acts on 2: to produce- this rotation. _ ' 25. Prove: 3'19 = 3491444), A: = 0,:1, :2, ' ' emflk‘“ =' cos (a + 2kr)-+ i sin (07+ 2hr) = coes +zi sin 9 V: 3"" 26; Evaluate each of the following. (a) [3(cos 40.0 + isin 40°)][4(cos 30° + :1 cm so°)] it ii .3 Eater + Meme-*6) J} Gems-*9)? +__4(_ei9)(e*i0)3 + (fa—W} I ( e‘zio’+ (2‘9 2 = -12<—%+€4) = —6 + 6V§4 (2 ci515°)7 _ 1281:1510? =- ' o_'1'3 o ‘ ' 7 (b ) (4 cis45°J3 " 64ci3135° V 243(105 5 i _ . =7 amt—30°) + isin(—30_°)} =' 2[cos30° -£sin 30°} = V§ __ 4 1+\/§£ 1" 2cis£60°)" ’0 . . _- =___ z '120102 100°: 120°: (c)_ (-1 _ Vii) 2 cis (—6”) (c15- 0 ) cm 2 (:13 Another method. ' _ ' ' _ . 10 . M - = ( Benji.3 >10: = (ezfil3)lo _= 82011173 1 -— 1/5 1' ' 23“”3 . \f— «.=' amaze/3 '= ‘(Ificos (2,13) + r13.”,inr(2az-l3)]r"—‘ w; + .2—3 v3. 'NIH 3 - 4403010“ + 80‘?) + i sin ‘(40° +' 809)] ' 12(405 120° + i sin 120°) “Io ...
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