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