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2007 12 ! # $ % &’ ()*+ , -. / ’01 2 (1 19)3 4 5 6 7 89 :; =Æ! ?@ A! B CD # E F $G H8IJK %LM2007 12 18 NO1.11.(1)i=cosπ2+isinπ2=eiπ2,(2)−1=cosπ+isinπ=eiπ(3)ρ=2,tanϕ=−√3, ϕ=2π3−1+i√3=2(cos2π3+isin2π3)=2ei2π3(4)1−i1+i=(1−i)22=−i=cos3π2+isin3π2=ei3π22.(1)3√i=3√eiπ2=eiπ2+2kπ3=ei(π6+2kπ3),(k=0,1,2,)(2)a+bia−bi=(a+bi)2a2+b2=a2−b2a2+b2+i2aba2+b23. 14. z0 R P 5. z=reiθ,zρeiϕ=rρei(ϕ+θ), ρeiϕ z z (ρ1 ) (ρ1 )ρ Q(ρ0) Q(ρ0) Rϕ 6.z4=−1=eiπ,Æzk=eiπ+2kπ4(k=0,1,2,3).z0=eiπ4 z1=ei3π4 7. |v|=√2, argv=3π4.8. 2 Æ Æ |z1−z2|≥||z1|−|z2||,|z1+z2|≤|z1|+|z2|NO1.31. C−R 12x+ay=dx+2y,ax+2by=−2cx−dy. x,y S a=d=2,b=c=−1,T u,v ! !a=d=2,b=c=−1 f(z) 2.|f(z)|2=u2x+v2x=uxvy−vxuy=uxuyvxvy.3. f(z)=u+iv(1) D f(z)≡0,ux+ivx≡0,ux=vx≡0 vy−iuy≡0,uy=vy≡0,#$u(x,y)=c1,v(x,y)=c2,(c1,c2 % ),f(z)=c1+ic2=c(% ).(2) Ref(z)=c, ux=uy=0,#$vy=vx=0,v=˜c f(z)=c+i˜c % U Imf(z)=cUf(z) % (3)|f(z)|=u2+v2=c, x y & 2uux+2vvx=0,2uuy+2vvy=0, C−R uux−vuy=0,uuy+vux=0, u ’ v ’( (u2+v2)ux=0,ux=0( u,v 0); v ’ u ’( uy=0,#$Uu=c, f(z) % 4.(1)f(z)=Rez=x,ux=1=vy=0, (2)f(z)=|z|=√x2+y2,ux=x√x2+y2=vy=0(!z=0 ), z=0,fz=|z|z! z=0,f(z) & 5.ux=vy,vx=−uy, x=ρcosϕ,y=ρsinϕ, uρ=uxcosϕ+uysinϕ,vϕ=vx(−ρsinϕ)+vyρcosϕ=ρ(uysinϕ+uxcosϕ)=ρuρ,(1)uϕ=ux(−ρsinϕ)+uyρcosϕ,vρ=vxcosϕ+vysinϕ=−uycosϕ+uxsinϕ=−1ρuϕ,(2) (1),(2) V)#$ C−R uρ=1ρuϕ,vρ=−1ρuϕ6.(%W V)#$ * uxx+uyy=0, x=ρcosϕ,y=ρsinϕ, ρ=√x2+y2,ϕ=arctanyx,ux=uρxρ+uϕ(−yρ2),uy=uρyρ+uϕxρ2,2uxx=uρρx2ρ2+uρ(1ρ−x2ρ3)+uϕϕy2ρ4+uϕ2yxρ4,uyy=uρρy2ρ2+uρ(1ρ−y2ρ3)+uϕϕx2ρ4+uϕ·−2yxρ4,uxx+uyy=uρρ+1ρuρ+1ρ2uϕϕ#$ X V)#$ * uρρ+1ρuρ+1ρ2uϕϕ=07.u&W ( !& uxx+uyy=2−2=0, + ’, v, dv=vxdx+vydy=−uydx+uxdy=(−2x+2y)dx+(2x+2y)dy,v=x0−2xdx+y0(2x+2y)dy+c=−x2+2xy+y2+c,f(z)=u+iv=(x2+2xy−y2)+i(−x2+2xy+y2+c)=(1−i)z2+ic8.(1) V)u=1ρ2cos2ϕ,vρ=−1ρuϕ=2ρ3sin2ϕ,vϕ=ρuϕ=−2ρ2cos2ϕ,dv=2ρ3sin2ϕdρ−2ρ2cos2ϕdϕ=d(−1ρ2sin2ϕ)v=−1ρ2sin2ϕ+c,f(z)=1ρ2(cos2ϕ−isin2ϕ)+ic=1ρ2e−i2ϕ+ic=1z2+ic, f(∞)=0,Uc=0, f(z)=z−2.- .(/ W V) v=−2xy(x2+y2)2+c,( f(z)=1z2.(2)du=uxdx+uydy=vydx−vxdy=12dx2−12dy2−2d(xy),u=12x2−12y2−2xy+c,f(z)=u+iv+c=iz2+12z2+c, f(0)=0,Uc=0. f(z)=(12+i)z2.NO1.431.(1)|ez2|=|ex2−y2+2ixy|=ex2−y2.(2)Ln(−1)=i(π+2kπ)=(2k+1)πi(k∈Z)(3)(1+i)i=eiLn(1+i)=ei[ln√2+i(π4+2kπ)]=eiln√2·e−(π4+2kπ).(4) i√i=z, Lnz=1iLni=1i·i(π2+2kπ)=π2+2kπ, z=eπ2+2kπ.(5)sin(a+ib)=sinacosib+cosasinib=sinacoshz+icosasinhb,cos(a+ib)=cosacosib−sinasinib=cosacoshb−isinasinhb,2.(1)ez=−1,z=Ln(−1)=i(π+2kπ)=i(2k+1)π.(2)sinz=2,eiz−e−iz=4i,e2iz−4ieiz−1=0,eiz=2i±√−3=(2±√3)i,iz=Ln(2±√3)i=ln(2±√3)+i(π2+2kπ),z=(12+2k)π−iln(2±√3)3.(1) z=x+iy,ez=ex(cosy+isiny)=ex(cosy−isiny)=ex−iy=ez(2)sinz=(eiz−e−iz2i)∗=e−iz−eiz−2i=eiz−e−iz2i=sinz(3)cosz=(eiz+e−iz2)∗=e−iz+eiz2cosz4.(1) z=ρeiϕ,zn=ρn(cosnϕ+isinnϕ),#,u=ρncosnϕ)’,v=ρnsinnϕ*$ & uρ=nρn−1cosnϕ=1ρvϕ,vρ=−1ρuϕ, zn +X(zn)=nzn−1 . 0# (xn)=nxn−1 X. .( f(z)=ux+ivx=(uρρx+uϕϕx)+i(vρρx+vϕϕx)=[uρcosϕ+uϕ(−sinϕϕ)]+i[vρcosϕ+vϕ(−sinϕϕ)] (zn)=[nρn−1cosnϕcosϕ+nρn−1sinnϕsinϕ]+i[nρn−1sinnϕcosϕ−nρn−1cosnϕsinϕ]=nρn−1[cos(n−1)ϕ+isin(n−1)ϕ]=nzn−1 . , %. n=1 z=1=1·z01 n=k &(zk)=kzk−1, (zk+1)=(zzk)=zzk+z(zk)=zk+z·kzk−1=(k+1)zk &Y$ &(zn)=nzn−1 (2)ez=ex(cosϕ+isinϕ),u=excosϕ,v=exsinϕ ux+ivx=excosϕ+iexsinϕ=ez, (ez)=ez4NO1.51.- ’f(z)=w(z)=a, −→f=a−→i.( ϕ=Reaz=ax,1(.x=c.) ψ=Imaz=ay,).y=c c # 2. t=x2+y2, txx+tyy=4=0,t *+ +Z ( #,u, u=F(t) + uxx+uyy=(Ft2x+Ftxx)+(Ft2y+Ftyy)=4F(x2+y2)+4F=4Ft+4F=0,FF=−1t,F=c1t,F=c1lnt+c2, u=c1ln(x2+y2)+c2.dv=−uydx+uxdy=−2c1yx2+y2dx+2c1xx2+y2dy,v=2c1arctanyx+c3,( −2c1arctanxy+c3). (f(z)=u+iv=c1ln(x2+y2)+c2+i(2c1arctanyx+c3)=2c1(ln|z|+iargz)+c2+ic3=2c1lnz+c2+ic3,(c1,c2,c3 # ).3. w=u+iv,w2=u2−v2+2iuv=z=x+iy,u2−v2=x,2uv=y,#$y2=4u2v2=4u2(u2−x),y2=4v2(v2+x), u=c1, 23. *y2=4c21(c21−x) 4 ,/.[ \5 v=c2, 1(. *y2=4c22(c22+x),4 ,/.[ 0 ( z=ρeiϕ, w=±√ρeiϕ223. √ρcosϕ2=c1,ρ=2c211+cosϕ ,/.1(. √ρsinϕ2=c2,ρ=2c221−cosϕ( ,/.4. . f(z)=i2σ(lnR−lnz)=i2σ(lnR−ln|z|−iargz)( v=Imf(z)=2σlnR−2σln√x2+y2=2σlnR−σln(x2+y2)3-−→E=−∇v=−(vx−→i+vy−→j)=2σxx2+y2−→i+2σyx2+y2−→j,E=|−→E|=2σ√x2+y2=2σρ .6 ES=qε0, 2σρ·2πρ=2πRτε0,]τ ^ 2 / τ=2σε0R. .( 0 u=Ref(z)=2σargz=2σϕ. ^ 2 / τ,71 ^ 22’ N, N=du=20π2σdϕ=4πσ. .6 N=2πRτε0, τ=2σε0R.NO2.21.(1)c:y=x=t,t#0 1 cRezdz=10t(1+i)dt=12(1+i)(2)#z=0 z=1,8 * x=t,y=0,t#0 1;#z=1 z=1+i,8 * x=1,y=t,t#0 15cRezdz=10tdt+10idt=12+i2.(1)|z|=1(z−2)dz=0.(2)|z−1|=41(z−1)2dz=0.(3)|z|=4(1z+i+2z−3+3z−5)dz=2πi+2×2πi+0=6πi.(4)−1−iisinzdz=−cosz|−1−ii=cosi−cos(1+i)=cosh1(1−cos1)+isin1sinh1.3.13 _ 4c ‘ 5 6 6 92 :7 437 ‘ 4 4.X c D z0 5; z1 a; c !76. c)c 8.c1 56 0=c1f(z)dz=c+c−f(z)dz, cf(z)dz=cf(z)dzT97cf(z)dz b)8 !# $ c 5;)a;$6 %&9 27] D 0|z|2 ’c1 |z|=1 P c2 $ P 5; −1 a; 1c11zdz=0πe−iϕieiϕdϕ=−πi;c21zdz=2ππidϕ=πi )8 &# NO2.31.(1)csinzzdz=2πi·sin0=0.(2)12πi|z−1|=4z2(z+4)2dz=0.(3)|z|=2cosπz(z−1)5dz