《复变函数论 Theory of Complex Variable Functions》课程授课教案（教材讲义，复分析 Complex Analysis）Chapter Ⅰ Complex Number Field

Chapter JComplex Number FieldInthischapter,wesurveythealgebraic andgeometric structureof thecomplexnumber systemWeassumevarious correspondingproperties ofreal numbers to beknown.Thepositive integernumbersystem,integernumbersystem,rationalnumbersystemandrealnumbersystemaredenoted by N,Z,Q and R, respectively.$1.1.SumsandProducts1.Definiton of Complex NumbersA complex mumber is defined as an ordered pair (x,y) of real numbers x and yIt is customaryto denoteacomplex number(x,y)by z,so that(1.1.1)z =(x,y)The real numbers xand y arecalled thereal and imaginaryparts of =,respectively,andwewriteRez=x,Imz=y(1.1.2)Two complex numbers z, =(x,yi) and z2 =(x2,y2) are equal whenever they have thesame real parts and the same imaginary parts.2. Operations of Complex NumbersThe sum z+z2 and the product zi2 of two complex numbers z=(x,yi) and22=(x2,y2）aredefinedasfollows(1.1.3)(x,y1)+(x2,y2)=(x, +X2,/ +y2),(1.1.4)(X,y)(x2,2) =(xix2 -yiy2, Jix2 +Xiy2)3. The Relationship of Real Numbers and Complex NumbersNote that the operations defined by equations (1.1.3) and (1.1.4) become the usual operations ofaddition and multiplication when restricted to the real numbers:(x,0)+(x2,0)=(x +x2,0), (x,0)(x2,0)=(xx2,0)The complex number system is, therefore, a natural extension of the real number system.4.Alternative Representation ofComplex Numbersy4Any complex number z=(x,y) can be written z=(x,y)as z =(x,0)+(O,y), and it is easy to see that(0,1)(y,0) = (0, y) . Hencei=(0,1)z =(x,0)+ (0,1)(y,0);+0x=(x,0)and, if we think of a real number x as the complexnumber (x,O),thatis,weidentifyareal numberxwith a correspondingcomplexnumber (x,O),and letFig. 1-1

Chapter Ⅰ Complex Number Field In this chapter, we survey the algebraic and geometric structure of the complex number system. We assume various corresponding properties of real numbers to be known. The positive integer number system, integer number system, rational number system and real number system are denoted by QZ,N, and R , respectively. §1.1. Sums and Products 1. Definiton of Complex Numbers A complex number is defined as an ordered pair yx ),( of real numbers x and y . It is customary to denote a complex number yx ),( by z , so that = yxz ),( . (1.1.1) The real numbers x and y are called the real and imaginary parts of , respectively; and we write z = Im,Re = yzxz . (1.1.2) Two complex numbers ),( and 111 = yxz ),( 222 = yxz are equal whenever they have the same real parts and the same imaginary parts. 2. Operations of Complex Numbers The sum and the product of two complex numbers and are defined as follows: 21 + zz 21zz ),( 111 = yxz ),( 222 = yxz ),(),(),( 2211 2121 + = + + yyxxyxyx , (1.1.3) (),)(,( , ) 2211 21212121 = − + yxxyyyxxyxyx . (1.1.4) 3. The Relationship of Real Numbers and Complex Numbers Note that the operations defined by equations (1.1.3) and (1.1.4) become the usual operations of addition and multiplication when restricted to the real numbers: )0,()0,()0,( 1 2 21 =+ + xxxx , )0,()0,)(0,( 21 21 = xxxx . The complex number system is, therefore, a natural extension of the real number system. 4. Alternative Representation of Complex Numbers Fig. 1-1 Any complex number can be written as , and it is easy to see that . Hence = yxz ),( += yxz ),0()0,( = yy ),0()0,)(1,0( xz += y )0,)(1,0()0,( ; and, if we think of a real number x as the complex number x )0,( , that is, we identify a real number x with a corresponding complex number x )0,( , and let

idenotetheimaginarynumber(O,l)(Fig.1-1)it isclearthatz=x+iy,(1.1.5)which iscalledtherectangularformof thenumberz.Thus,thecomplexnumber systemcanbewritten asC=((x,y):x,yeR)=(x+iy:x,yeR)22= zz,z”=zz,etc.,wefind thatAlso, with the convention2 = (0,1)(0,1) = (-1,0) = -1(1.1.6)Thus,the equation z? +1=0 has a root z=i in CIn view ofexpression (1.1.5), definitions (1.1.3) and (1.1.4) become(1.1.7)(x, +iyi)+(x2 +iy2)=(xi +x2)+i(yi+y2),(1.1.8)(x+iy(x+iy)=(xx-yy)+iyxz+xy)Observe that the right-hand sides of these equations can be obtained by formally manipulating theterms ontheleft replacing?2by-l when itoccurs

i denote the imaginary number )1,0( ( Fig. 1-1) it is clear that z = x + iy , (1.1.5) which is called the rectangular form of the number . Thus, the complex number system can be written as z C = ∈R = + yxiyxyxyx ∈R},:{},:),{( . Also, with the convention , etc., we find that 2 23 , == zzzzzz 1)0,1()1,0)(1,0( 2 i = −=−= . (1.1.6) Thus, the equation 01 has a root 2 z =+ = iz in C . In view of expression (1.1.5), definitions (1.1.3) and (1.1.4) become )()()()( 2211 21 21 + + + = + + + yyixxiyxiyx , (1.1.7) ())(( () ) 2211 2121 2121 + + = − + + yxxyiyyxxiyxiyx . (1.1.8) Observe that the right-hand sides of these equations can be obtained by formally manipulating the terms on the left replacing by -1 when it occurs. 2 i

Complex Number FieldIChapterIs1.2.BasicAlgebraicPropertiesVarious properties of addition and multiplication of complexnumbers are the same as forrealnumbers. We list here the more basic of these algebraic properties and verify some of them. Mostof the others are verified in the exercises.1.Commutativelaw(1.2.1)2,+22=22+212172=22712.Associative law3.Distributivelaw2(2, +z2) = 22, + 272,(1.2.3)nz=z+z+..+z and z" =zz...z4.IdentitiesTheadditiveidentity0=(0,0)and the multiplicative identity1=(10)for real numberscarry over to the entire complex number system. That is,z+0=z and z-1=z(1.2.4)foreverycomplexnumberz.Furthermore,O and1aretheonlycomplexnumberswithsuchproperties (see Exercise 9).5.Additive inverseFor eachcomplex number z =(x,y),there is an additive inverse(1.2.5)-z =(-x,-y),6.Subtraction21 -22 =z,+(-22), Vz1,32 eC.(1.2.6)So if z, =(x,y) andz,=(x2,y2),then(1.2.7)z, -z2 =(x -x2,yi - y2)=(x, -x2)+i(y1 - y2)7.Multiplicative inverseFor z=(x,y)=x+iy±, theres a numbersuch that=1,called theisZzItisfindthat theofmultiplicative inverseofZeasytomultiplicativeinversez=(x,y)=x+iy isxx-y-yZ(= + 0),+i(1.2.8)x+ y2x+y2x+yx+yFrom the discussion above, we conclude that the set C of all complex numbers becomes afield,calledthefieldofcomplexnumbers,orthecomplexnumberfield

Chapter Complex Number Field Ⅰ 1 §1.2. Basic Algebraic Properties Various properties of addition and multiplication of complex numbers are the same as for real numbers. We list here the more basic of these algebraic properties and verify some of them. Most of the others are verified in the exercises. 1. Commutative law 12211221+ = + , = zzzzzzzz (1.2.1) 2. Associative law 3. Distributive law 21 21 + )( = + zzzzzzz , (1.2.3) 6474 484 L n +++= zzznz and 876 L n n = zzzz . 4. Identities The additive identity = )0,0(0 and the multiplicative identity = )0,1(1 for real numbers carry over to the entire complex number system. That is, + 0 = zz and ⋅1 = zz (1.2.4) for every complex number z . Furthermore, 0 and 1 are the only complex numbers with such properties (see Exercise 9). 5. Additive inverse For each complex number = yxz ),( , there is an additive inverse − = − −yxz ),( , (1.2.5) 6. Subtraction − = + − 2121 ∀ ,),( zzzzzz 21 ∈C . (1.2.6) So if ),( and , then 111 = yxz ),( 222 = yxz )()(),( 212121 21 21 =− − − = − + − yyixxyyxxzz . (1.2.7) 7. Multiplicative inverse For ),( == + iyxyxz ≠ 0 , there is a number such that , called the multiplicative inverse of . It is easy to find that the multiplicative inverse of is −1 z 1 1 = − zz z ),( +== iyxyxz , )0( 2222 2222 1 ≠ + − + + =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + = − z yx y i yx x yx y yx x z . (1.2.8) From the discussion above, we conclude that the set of all complex numbers becomes a field, called the field of complex numbers, or the complex number field. C

$1.3.FurtherPropertiesInthissectionwementionanumberofotheralgebraicpropertiesofadditionandmultiplicationof complexnumbersthatfollowfromtheonesalreadydescribedinSec.1.2.Becausesuchproperties continue to be anticipated, the reader can easily pass to Sec.l.4 without seriousdisruption.1.Expunctive lawIf z=,=0,then either z,=0 or z2=0; or possibly both z and =2 equal zero.Another way to state this result is that if two complex numbers z and z2 are nonzero, then sois their product z,-2.2.DivisionDivisionbyanonzerocomplexnumber isdefinedasfollows:=222(z, ±0)(1.3.1)Z2If z,=(x,)= x, +iy) and z, =(x2,y2)=x, +iy2,then= +yy2+iy-xy2(=2±0)(1.3.2)x,+ysx2+y2Z2Although expression (1.3.2) is not easy to remember, it can be obtained by writing (see Exercise4)三_ (x +iy)(xz-iy2)(1.3.3)22(x2 +iy2)(x2 -iy2)3.Useful identities=2(=2 ± 0)(1.3.4)22(2, 0),(1.3.5)2222(2/22)(z2)=(2-7")(2232) =1 (22 ±0)()=(2,2) =27(z ± 0,z2 ± 0)(1.3.6)2,222122Z122(z3 ± 0,=4 ± 0),(1.3.7)23-423Z4Example.Computations such as thefollowing are now justified:5+i115+i1112-3i1+i)=5-i5+i(5-i)(5+i)(2 -3i)(1+i)5+i_5-5.1i26262626264.Binomial formulaIf z, and z, are any two complex numbers, then(n)n-(z +z,)">(n = 1,2,...)(Binomial Formula)(1.3.8)22Kk=o(k)n!(k = 0,1,2,..,n) and where it is agreed that Ol=1wherek!(n-k)!

§1.3. Further Properties In this section, we mention a number of other algebraic properties of addition and multiplication of complex numbers that follow from the ones already described in Sec.1.2. Because such properties continue to be anticipated, the reader can easily pass to Sec.1.4 without serious disruption. 1. Expunctive law If 0 , then either or zz 21 = 0 z1 = 0 z2 = ; or possibly both and equal zero. Another way to state this result is that if two complex numbers and are nonzero, then so is their product . 1 z 2 z 1 z 2 z 21 zz 2. Division Division by a nonzero complex number is defined as follows: )0( 2 1 21 2 1 = ≠ − zzz z z (1.3.1) If and 11111 ),( +== iyxyxz 22222 = ),( = + iyxyxz , then )0( 2 2 2 2 2 2121 2 2 2 2 2121 2 1 ≠ + − + + + = z yx yxxy i yx yyxx z z . (1.3.2) Although expression (1.3.2) is not easy to remember, it can be obtained by writing (see Exercise 4) ))(( ))(( 2222 2211 2 1 iyxiyx iyxiyx z z −+ −+ = . (1.3.3) 3. Useful identities )0( 1 2 1 2 2 = ≠ − zz z . (1.3.4) )0( 1 2 2 1 2 1 ≠ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = z z z z z . (1.3.5) )0(1))(())(( 2 1 22 1 11 1 2 1 121 = ≠= −− − − zzzzzzzzz . )0,0( 11 )( 1 1 2 21 1 2 1 1 1 21 21 ≠≠ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ === −−− zz zz zzzz zz . (1.3.6) )0,0( 3 4 4 2 3 1 43 21 ≠≠ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = zz z z z z zz zz . (1.3.7) Example. Computations such as the following are now justified: . 26 1 26 5 2626 5 26 5 )5)(5( 5 5 5 5 1 )1)(32( 1 1 1 32 1 i ii ii i i i iiiii +=+= + = +− + = + + ⋅ − = +− ⎟ = ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − 4. Binomial formula If and are any two complex numbers, then 1 z 2 z ∑= − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =+ n k n kkn nzz k n zz 0 21 21 )( K),2,1( (Binomial Formula) (1.3.8) where ),2,1,0( )!(! ! k n knk n k n = K − =⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ and where it is agreed that = 1!0

$1.4.ModuliIt is natural to associateany nonzero complexnumber z=x+iywiththe directed line segmentor vector,from the origintothe point (x,y)thatrepresents z(Sec.1.1)in the complexplaneIn fact, we often refer to zas the point zor the vector z,In Fig.1-2,the numberz = x+ iy and - 2+i are displayed graphically as both points and radius vectors.J4(-2,1)人2+ix+iy(x,y)xo-2Fig. 1-2Accordingtothedefinition of thesumof two complexnumbersz,=x,+iy,andz2 = x, +iy2, z, + z2 may be obtained vectorially as shown in Fig. 1-3.The differencez1-=2 = z, +(-z2) corresponds to the sum of the vectors z and -z2(Fig. 1-4).yAy4(x2y)(xy)ZX8x02,-2Fig. 1-4Fig. 1-31.ModulusThe modulus, or absolute value, of a complex number z = x+iy is defined as thenonnegative real number x? +y?and is denoted by I=l; that is,[==/x? +y2(1.4.1)Geometrically,thenumber=isthedistancebetweenthepoint (x,y)andtheorigin,orthelength of the vector representing z: It reduces to the usual absolute value in the real numbersystemwheny=o.2.DistanceofcomplexnumbersThe distance between two points =, = x, +iy, and ≥, = x, +iy2 is defined by

§1.4. Moduli It is natural to associate any nonzero complex number z = x + iy with the directed line segment, or vector, from the origin to the point yx ),( that represents z (Sec. 1.1) in the complex plane. In fact, we often refer to z as the point z or the vector z . In Fig. 1-2, the number z x += iy and − 2 + i are displayed graphically as both points and radius vectors. Fig. 1-2 According to the definition of the sum of two complex numbers and , 1 1 1 z = x + iy 2 2 2 z = x + iy 1 2 z + z may be obtained vectorially as shown in Fig. 1-3. The difference ( ) 1 2 1 2 z − z = z + −z corresponds to the sum of the vectors and (Fig. 1-4). 1 z 2 − z 1. Modulus The modulus, or absolute value, of a complex number z = x + iy is defined as the nonnegative real number 2 2 x + y and is denoted by | z |; that is, 2 2 | z |= x + y . (1.4.1) Geometrically, the number is the distance between the point and the origin, or the length of the vector representing | z | (x, y) z . It reduces to the usual absolute value in the real number system when y = 0 . 2.Distance of complex numbers The distance between two points 1 1 1 z = x + iy and 2 2 2 z = x + iy is defined by

Iz, -z2 = /(x, -x) +(1 - y2)2The complex numbers = corresponding to the points lying on the circle with center Zoand radius R thus satisfy the equation Iz-zo = R, and conversely. We refer to this set ofthesepointssimplyasthecirclez-zR,denotedbyC(o,R)Example 2.The equation =-1+3i=2 represents the circle whose center is the pointZo =(1,-3) and whose radius is R = 2 .3.RelationshpsofzRezandImz[== (Re2)2 +(Im2)2(1.4.2)RezRez= and ImzImz=](1.4.3)4.Triangle inequality[2, ±z2 2, /+/22 1,(1.4.4)(1.4.5)12, ±22 [2, /-122 ]Example3.Ifapoint z lies on theunitcirclez-1 abouttheorigin,then[z - 2=/ +2 = 3and[2 -2|=]-2|=1.The triangle inequality (1.4.4) can be generalized by means of mathematical induction tosums involving any finite number of terms:(1.46)[2 +z2 ++zn 2, /+|22 /+.+[2, / (n=2,3,...)

2 21 2 21 21 −+−=− yyxxzz )()(|| . The complex numbers corresponding to the points lying on the circle with center and radius z 0 z R thus satisfy the equation − || = Rzz 0 , and conversely. We refer to this set of these points simply as the circle − || = Rzz 0 , denoted by ),( . 0 RzC Example 2. The equation − + iz = 2|31| represents the circle whose center is the point )3,1( and whose radius is z0 −= R = 2 . 3.Relationshps of z, Re z and Im z 2 2 2 += zzz )(Im)(Re|| . (1.4.2) ≤ ≤ zzz |||Re|Re and ≤ ≤ zzz |||Im|Im . (1.4.3) 4.Triangle inequality |||||| 121 2 ± ≤ + zzzz , (1.4.4) |||||| 2121 −≥± zzzz . (1.4.5) Example 3. If a point z lies on the unit circle z = 1|| about the origin, then − ≤ zz + = 32|||2| and zz =−≥− 12|||2| . The triangle inequality (1.4.4) can be generalized by means of mathematical induction to sums involving any finite number of terms: | ),3,2(||||||| + 21 L++ n ≤ 1 + 2 +L+ n nzzzzzz = K . (1.46)

s1.5.Conjugates1.Conjugate of a complex numberTheconjugateofacomplexnumberz=x+iyisdefinedasthecomplexnumber x-iy andis denoted by =; that is,2=x-iy.(1.5.1)The number = is represented by the point (x,-y), which is the reflection in the real axis of thepoint (x,y) representing z (Fig. 1-5).4(x,y)Z0xz+ (x, -y)Fig. 1-52.Useful identities2=212H1(1.5.2) + z2 = z) + Z2(1.5.3)21 -22 = 21 -22 (1.5.4)2/22 = 2/22,((z2# 0)(1.5.5)22)Z2Z+22-2Rez:Imz:(1.5.6)22iz2±≥2.(1.5.7)Example1.Asanillustration,wecompute)_-5+5i5+5i-1+ 3i _ (-1+ 3i)(2 + i)-l+i2-i12-i25(2 -i)(2 +i)Seealsotheexampleneartheendof Sec.1.3Identity (1.5.7)is especially useful inobtaining properties ofmoduli fromproperties ofconjugates noted above.We mention that[2)22 -[2, / 22 ](1.5.8)and1z1z, ± 0)(1.5.9)z.Z2Property (1.5.8) can be established by writing[2(22 /P= (2)22)(2,22) =(2)22)(2,2) =(2,=)(z2=2) =z, / =2 /P= (I =) ll 2 D)2and recalling that a modulus is never negative. Property (1.5.9) can be verified in a similar way

§1.5. Conjugates 1.Conjugate of a complex number The conjugate of a complex number z = x + iy is defined as the complex number x − iy and is denoted by z ; that is, −= iyxz . (1.5.1) The number z is represented by the point −yx ),( , which is the reflection in the real axis of the point yx ),( representing z (Fig. 1-5). 2. Useful identities Fig. 1-5 z = z , | z |=| z | 1 2 1 2 z + z = z + z . (1.5.2) 1 2 1 2 z − z = z − z , (1.5.3) 1 2 1 2 z z = z z , (1.5.4) ( 0) 2 2 1 2 1 = ≠ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ z z z z z . (1.5.5) i z z z z z z 2 , Im 2 Re − = + = . (1.5.6) 2 zz =| z | . (1.5.7) Example 1. As an illustration, we compute i i i i i i i i i i = − + − + = − − + = − + − + + = − − + 1 5 5 5 | 2 | 5 5 (2 )(2 ) ( 1 3 )(2 ) 2 1 3 2 . See also the example near the end of Sec.1.3. Identity (1.5.7) is especially useful in obtaining properties of moduli from properties of conjugates noted above. We mention that | | | || | 1 2 1 2 z z = z z (1.5.8) and ( 0) | | | | 2 2 1 2 1 = z ≠ z z z z . (1.5.9) Property (1.5.8) can be established by writing 2 1 2 2 2 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 2 | z z | = (z z )(z z ) = (z z )(z z ) = (z z )(z z ) =| z | | z | = (| z || z |) and recalling that a modulus is never negative. Property (1.5.9) can be verified in a similar way

Example 2. Property (1.5.8) tells us that [z}==} and I===β. Hence if zisapoint inside the circle centered at the origin with radius 2, so that I=k2, it follows from thegeneralized form (1.4.9) of the triangle inequality in Sec. 1.4 that12* +322 2z +1=P +3|= +2|≥/+1<25

Example 2. Property (1.5.8) tells us that and . Hence if is a point inside the circle centered at the origin with radius 2, so that 22 = zz |||| 33 = zz |||| z z < 2|| , it follows from the generalized form (1.4.9) of the triangle inequality in Sec. 1.4 that 251||2||3|||123| 23 3 2 zzzzzz <+++≤+−+

s1.6.ExponentialFormLet r and be polar coordinates of the point (x,y) that corresponds to a nonzero complexnumber z=x+iy: Since x=rcoso and y=rsino, the number z can be written inpolarformasz=r(cosの+isinの)(1.6.1)If z= 0, the coordinate is undefined, and so it is always understood that z + 0 wheneverargz or Argz defined below is discussed.In complex analysis, the real number r isnotallowed to benegativeand is the lengthof theradiusx+iyvector for z,thatis,r=z.Thereal number represents the angle,measured in radians, that zmakeswiththepositiverealaxiswhenzisinterpreted as a radius vector, see Fig.1-6.xAs in calculus, has an infinitenumber ofpossible values, including negative ones, that differ byFig. 1-6integralmultiplesof2元.Thosevaluescanbedeterminedfromtheequationtane=y/x,where the quadrant containing the argument of =, and the set of all such values is denoted byArgz. The principal value of Argz, denoted by argz, is the unique value such that-元<①≤元.NotethatArgz = [argz + 2nπ : n = 0,±1,±2,..],simply,wewriteArgz = argz +2n元(n =0,±1,±2..)(1.6.2)Also,whenzisanegativerealnumber,argzhasvalue元,not -元Example 1. The complex number -1-i, which lies in the third quadrant, has principal3元argument -3元 /4. That is, arg(-1-i) = -4It must be emphasized that, because of the restriction -π<≤ π of the principalargument @, it is not true that arg(-1-i)= 5元 /4.According to equation (1.6.2),we have3元+2n元(n=0,±1,±2,...).Arg(-1-i) = -4Note that the term argz on the right-hand side of equation (1.6.2) can be replaced by anyparticular value of Arg and that one can write, for instance,5元 +2m (n=0,±1±2..)Arg(-1-i) :4The symbol eie, or exp(io), is defined by means of Euler's formula aseie=cos+isino,(1.6.3)where is to be measured in radians. It enables us to write the polar form (1.6.1) morecompactlyinexponential formas2=reio(1.6.4)The choice of the symbol eio will be fully motivated later on is Sec.2.8. Its use in Sec. 1.7 will,however, suggest that it is a natural choice.Example2.Thenumber-1-iinExample1hasexponentialform

§1.6. Exponential Form Let r and θ be polar coordinates of the point that corresponds to a nonzero complex number yx ),( z x += iy . Since = rx cosθ and = ry sinθ , the number can be written in polar form as z = θ + irz θ )sin(cos . (1.6.1) If z = 0 , the coordinate θ is undefined; and so it is always understood that whenever or defined below is discussed. z ≠ 0 arg z Argz Fig. 1-6 In complex analysis, the real number r is not allowed to be negative and is the length of the radius vector for z ; that is, r = z || . The real number θ represents the angle, measured in radians, that makes with the positive real axis when is interpreted as a radius vector, see Fig. 1-6. z z As in calculus, θ has an infinite number of possible values, including negative ones, that differ by integral multiples of 2π . Those values can be determined from the equation θ = /tan xy , where the quadrant containing the argument of , and the set of all such values is denoted by . The principal value of , denoted by , is the unique value such that z Argz Argz argz Θ π ≤Θ<− π . Note that = + π nnzz = ± ± K},2,1,0:2arg{Arg , simply, we write = + π nnzz = ± ± K),2,1,0(2argArg . (1.6.2) Also, when z is a negative real number, argz has value π , not −π . Example 1. The complex number −1− i , which lies in the third quadrant, has principal argument − π 4/3 . That is, 4 3 )1(arg π i −=−− . It must be emphasized that, because of the restriction −π < Θ ≤ π of the principal argument Θ , it is not true that − − i = π 4/5)1(arg . According to equation (1.6.2), we have ),2,1,0(2 4 3 )1(Arg ±±=π+ K π i −=−− nn . Note that the term on the right-hand side of equation (1.6.2) can be replaced by any particular value of and that one can write, for instance, argz Argz ),2,1,0(2 4 5 )1(Arg ±±=π+ K π =−− nni . The symbol , or iθ e iθ )exp( , is defined by means of Euler’s formula as θθ θ e isincos i += , (1.6.3) where θ is to be measured in radians. It enables us to write the polar form (1.6.1) more compactly in exponential form as iθ = rez (1.6.4) The choice of the symbol will be fully motivated later on is Sec.2.8. Its use in Sec. 1.7 will, however, suggest that it is a natural choice. iθ e Example 2. The number 1−− i in Example 1 has exponential form

-1-i= /2 expl(1.6.5)Since e-=el(-0), this can also be witten -1-i= 2e-3x/4, Expression (1.6.5) is, ofcourse,onlyoneofan infinitenumber of possibilitiesfor theexponential form of-1-i:3元+2m元-1-i=/2expl(n = 0,±1,±2,...).(1.6.6)4Note how expression (1.6.4) with r =1 tells us that the numbers eie lie on the circlecentered at the origin with radius unity, as shown in Fig. 1-7. Values of eieare, then, immediatefrom that figure, without reference to Euler's formula. It is,for instance,geometrically obviousthat ei" =-1,eix/2 =i, and e-14m =1.TRe60XFig. 1-7Fig.1-8Note,too,thattheequationZ= Re'(0≤0≤2元)(1.6.7)is a parametric representation ofthe circle = R, centered at the origin with radius R. As theparameter increases from =0 to =2元, the point = starts from the positive real axisand traverses the circle once in the counterclockwise direction More generally,the circleIz-z-R,whose center is zo and whose radius is R,has the parametric representationz= z。+ Re0(0≤0≤2元).(1.6.8)This can be seen vectorially (Fig. 1-8) by noting that a point z traversing the circle[z-z0 Ronce in the counterclockwise direction corresponds to the sum of the fixed vector -o and avectoroflength Rwhoseangleof inclination variesfrom=0 to =2元

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =−− − 4 3 exp21 π ii . (1.6.5) Since , this can also be written θ−θ− )( = ii ee 4/3 21 π− =−− i ei . Expression (1.6.5) is, of course, only one of an infinite number of possibilities for the exponential form of 1−− i : ),2,1,0(2 4 3 exp21 ⎥ ±±= K ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π+ π =−− − nnii . (1.6.6) Note how expression (1.6.4) with r = 1 tells us that the numbers lie on the circle centered at the origin with radius unity, as shown in Fig. 1-7. Values of are, then, immediate from that figure, without reference to Euler’s formula. It is, for instance, geometrically obvious that and . iθ e iθ e ,1 2/ ieei i =−= π π 1 14 = − i e π Fig. 1-7 Fig.1-8 Note, too, that the equation = (0 ≤ θ ≤ 2π) iθ z Re (1.6.7) is a parametric representation of the circle | z |= R , centered at the origin with radius R . As the parameter θ increases from θ = 0 to θ = 2π , the point starts from the positive real axis and traverses the circle once in the counterclockwise direction. More generally, the circle , whose center is and whose radius is z | z − z0 |= R 0 z R , has the parametric representation (0 2 ) = 0 + ≤ θ ≤ π iθ z z Re . (1.6.8) This can be seen vectorially (Fig. 1-8) by noting that a point z traversing the circle | z − z0 |= R once in the counterclockwise direction corresponds to the sum of the fixed vector and a vector of length 0 z R whose angle of inclination θ varies from θ = 0 to θ = 2π