0 Complex Numbers lz-ti=Rez+t -t Figure 1.3 Graphs for Example 2. ●2=a+bi ●=a-bi Figure 1.4 Complex conjugates. Complex Conjugates.The reflection of the point a =a+bi in the real axts is the point a-bi (see Fig.1.4).As we shall see,the relationship between a bi and a-bi will play a significant role in the theory of complex variables.We introduce special notationforthisconceptinthenextdeinition Definition 4.The complex conjugate of the number =a+bi is denoted by and is given by =a-bi Thus, -1+57=-1-5i,π-7=π+i,8=8. omruheastersdenote thecomplexconugate. It follows from Definition 4 that if and only if z is a real number.Also it of the sum(difference)of two complex numbers is equal to (difference)of their coniugates:that conjugate 1十2=+五，t-=方-2
1.2 Point Representation of Complex Numbers 11 Perhaps not so obvious is the analogous property for multiplication. Example 3 Prove that the conjugate of the product of two complex numbers isequalto the product of the conjugates of these numbers Solution. It is requlred to verify that (G12=石2 (2) Write zt =a +bii,z2 =t+bi.Then (132)=ataz -b1b2 (ajh +abn)i =a1a2-b1b2-(a1b2+a2b1)i. On tle other hand. i=(a1-b1i)(az-bi)=aia2-b1b2-abai-azbi ata2-b1b2-(atb +azb1)1. Thus Eq.(2)holds.■ In addition to Eq.(2)the following properties can be seen: (3) R@?=2+8 2 (4) ⊙ whcreas(5)shows that the difference is(pure)imaginary.The conjugate of the conju gate of a complex number is,of course,the number itself: It is clear from Definition 4 that lzl that is,the points z and are equidistant from the origin.Furthermore,since =(a+bi)(a-bi)=a2+b2, we have =lzl2 (7)
农 Conplex Numbers This is a useful fact to remember:The square of the modulus of a complex mumber als the wube s its conjugate. y we ha oyed complex conjugates in Sec.1.1.in the process enominator for the division algorithm.Thus,for inst ance,if z and are complex numbers,then we rewrite as a ratio with a real denominator by using 2: 鼎器漂 (8) In particular, 1 =印 (9) In closing we would like to mention that there is another,possibly more enlight ening,way to see Eq.(2).Notice that when we represent a c mber of two real numbers and the symbol i.as in z= +bi,then r in terms heationofcoajuege ihnhesigeNowatothatplaysin computations;it merely holds a pla by -I whenever it arises.Ex hiwemuteodnits the computations:we kcept for these occurrencesi is never really absorbed into square we a uld just as well call itj.,or any other syr e to replace by-1.In fact,without affecting the mbol who validiry of the caicula- replace it throughou by the symbol ()since the square of the larter y nd then multiply,the only thing diffe (a1+bii)(a2+b2f)we replace i rent about the product will be the a statement of Example 3. EXERCISES 1.2 1.Show that the point (+)/2 is the midpoint of the line scgment joining and 22. 21 mne cer ooy icles of masses 2,1.3,and 5 located at the respective points1+i. 3.Which of the pointsi,2-i,and-3 is farthest from the origin? 4.Let-1.Plot the poinsad in the complex plane.Do the same for=2+3i and z=-2i. 5.Show that andthe vertices of 6.Show that the points36,and 4+4 are the vertices of a right triangle. By the same lok ewe should be abie to repacebyn()her ne result.Try it.】
1.2 Point Representation of Complex Numbers 13 7.Describe the set of pointszin the complex plane that satisfieseach of the following. (a)Im z=-2 (b)z-1+=3 (c)2z-i1=4 (d)lz-l川=lz+i川 (e)z=Rez+2 (02-川+k+1川=7 (g)z=3z-I1 (h)Rez≥4 ()z-1<2 G0)z1>6 8.Show,both analytically and graphically,that-1=-1l. 9.Show that if r is a nonnegative real number,then ral=rl. l0.Prove that |Rez≤l and Im zl≤lzl, 11.Prove that if=Re,then zis a nonnegative real number. 12.Verify properties(3),(4),and(5). 13.Prove that if()2=22,then zis either real or pure imaginary. k.Ve Be ()ad ()ow 14.Prove 15.Prove that ()=(for every integer&(provided0 when is negative). 16.Prove that if=1(e≠1)，then Re[l/l-zl= 18.Use the familiar formula for the roots of a quadraric polynomial to give another proof of the statement in Prob.17 for the case n=2. 19.We have noted that the conjugate ()is the reflection of the point zin the real axis (the horizontal line y=0).Show that the reflection of z in the line ax+by =c (a,b.c real)is given by 2ic+(b-a)2 一0 rces i6geLc1 i be an m by分matr whoe ntc。 plex numbers we dcno tis obtained by formin the transpose (intercha ing rows ved by taking the conju gate of each enuy.In other words.if B-thenB [byE).E xample: [42]-[梦][]-1-
Complex Numbers 14 For an by matrix A=[with complex entries.prove the following: (a)If uAuforalbyIo vectorsuwith comple cntres,then A is 产 =0,take u (b)Show by example that the conelusion ("A is the zero matrix")can fail if the thesis for part (a)only holds for vectors u with read number entries NT:Try to find a2 by 2 real no matix A such that u'Au= all real by I vectorsu. 21.Let A be an mby n matrix with complex entries.We say that A is Hermtitean if At =A(see Prob.20). (a)Show that if A is Heitean,thenuAu is real o vecto u with complex entrics (e)Show that if B is any by n m rix and u is any by I column vector(each with complex entries).then uB'Bu must be a nonnegative real number. 1.3 Vectors and Polar Forms With each point in the complex plane we can associatea vector,namely,the directed line segment from the origin to the point Recall that vectors are characterized by length and direction.and that a given vector remains unchanged under translation. Thus the vecor det termined by z=1+i is the same as the vector fron n the point oint3+(sce 15).Note that every vector parallel to the real axis corresponds to a real number,while those parallel to the imaginary axis represent pure imaginary numbers.Observe,also,that the length of the vector associated with is Figure 15 Complex numbers as vectors