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Unformatted text preview: WPMES Mug _. 4 _ _. COMPLEXNUMBERS , ' [cane1 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— — xzH'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 212262,P = r1r2r,. {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
Vinthe 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) = cos2«/n—+isin 21r/1‘t gem», the a roots are 1, a), :92, ...,m““. Geo . metrically they represent the n vertices of aregular polygonof 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.'14. 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. 14, are considered equal. .
Hence we write 01’ = AB = a; +iy. 7 7 A _.Fig.14 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 (twodimensional) 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 31143: 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 nopointsat '
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 noncountable or nondenumemble. 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 HeineBore] 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)+(7i) = 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—1l2i} + (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) 8i4i—12i—6i‘a 8+4i—12i—l6
(g) (4+2i)(2—3i);= 4(2—31‘) +2i(2—3i) 8—12i+4i+6i2 8 —12i+4il?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 14 _ 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). = (00—11, o1_+ 10) = (—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: “lb3+ 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\/§£ . \ ' ' '
' Modulusor 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.‘IZz
’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° = 31214 (radians)
Then —5 +' 512 = 5V5 (cos 135° + isin 1359)
' ' = 5(2— cigar/4 = 5J5 chum (b). —5 + 5% 1. ' . + = I«é—mi = «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(cos3r/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. 126 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 stanzmitgzmmaami 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 461998} ) 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(e39  9‘3“ 2'5 _ 1  . 1" é4i°+074i9  1
o  = _. so 219 —2iﬂ —m = __ .____, ..
16(e4_+4e_l6+4e +e ) 8( 2 ),+_2 __ l ' l E — 8c0340+2cosz2ﬂ[P8 geometrically 28*“ Where a is real. Let 2:412“? be represented graphically by vector
0A in Fig. 1—30. Then zeia' ..—.. remgia_ — Mimic) 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(2azl3)]r"—‘ w; + .2—3 v3. 'NIH 3  4403010“ + 80‘?) + i sin ‘(40° +' 809)] '
12(405 120° + i sin 120°) “Io ...
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