数理方法习题答案

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1、 200712!"#$%&'()*+,-./'01(119)34678
:;<=>Æ! ?@A"! BCD#EF$GH8IK%LM20071218NO1.1ππiπ1.(1)i=cos+isin=e2,22(2)−1=cosπ+isinπ=eiπ√(3)ρ=2,tanϕ=−3,2π√ϕ=32π2πi2π−1+i3=2(cos+isin)

2、=2e3331−i(1−i)23π3πi3π(4)==−i=cos+isin=e21+i222√√π+2kπ33iπi2i(π+2kπ)2.(1)i=e2=e3=e63,(k=0,1,2,)a+bi(a+bi)2a2−b22ab(2)a−bi=a2+b2=a2+b2+ia2+b23.14.z0RP5.iθiϕi(ϕ+θ)iϕz=re,zρe=rρe,ρe
zz(ρ>1)(ρ<1)ρQ(ρ>0)Q(ρ<0)Rϕ4iπiπ+2kπ6.z=−1=e,Æzk=e4(

3、k=0,1,2,3).iπi3πz=e440z1√=e7.3π

4、v

5、=2,argv=4.8.2ÆÆ

6、z1−z2

7、≥

8、

9、z1

10、−

11、z2

12、

13、,

14、z1+z2

15、≤

16、z1

17、+

18、z2

19、NO1.31.C−R12x+ay=dx+2y,ax+2by=−2cx−dy.x,ySa=d=2,b=c=−1,Tu,v!!a=d=2,b=c=−1f(z) ""uu222xy2.

20、f(z)

21、=ux+vx=uxvy−vxuy=

22、.vxvy(z)≡0,u+iv≡0,u=v≡03.f(z)=u+iv(1)Dfxxxxvy−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=˜cf(z)=c+ic˜%UImf(z)=cUf(z)%(3)

23、f(z)

24、=u2+v2=c,xy&2uux+2vvx=0,2uuy+2vvy=0,C−Ruux−vuy=0,uuy+vux=

25、0,u'v'((u2+v2)u=0,u=0(xxu,v0);v'
u'(uy=0,$Uu=c,f(z)%4.(1)f(z)=Rez=x,ux=1=vy=0,√(2)f(z)=

26、z

27、=x2+y2,u=√x=v=0(f

28、z

29、xx2+y2y!z=0),z=0,z=z!"z=0,f(z)&5.ux=vy,vx=−uy,x=ρcosϕ,y=ρsinϕ,uρ=uxcosϕ+uysinϕ,vϕ=vx(−ρsinϕ)+vyρcosϕ=ρ(uysinϕ

30、+uxcosϕ)=ρuρ,(1)uϕ=ux(−ρsinϕ)+uyρcosϕ,1vρ=vxcosϕ+vysinϕ=−uycosϕ+uxsinϕ=−uϕ,(2)ρ(1),(2)V)#$C−R11uρ=uϕ,vρ=−uϕρρ6.(%WV)#$*uxx+uyy=0,√x=ρcosϕ,y=ρsinϕ,22yρ=x+y,ϕ=arctanx,xyyxux=uρρ+uϕ(−ρ2),uy=uρρ+uϕρ2,2x21x2y22yxuxx=uρρρ2+uρ(ρ−ρ3)+uϕϕρ4+uϕρ4,y21y2x2−2yxuyy=uρρρ2

31、+uρ(ρ−ρ3)+uϕϕρ4+uϕ·ρ4,11uxx+uyy=uρρ+uρ+2uϕϕρρ$XV)#$*11uρρ+uρ+uϕϕ=0ρρ27.u&W(!&uxx+uyy=2−2=0,+',v,dv=vxdx+vydy=−uydx+uxdy=(−2x+2y)dx+(2x+2y)dy,xy22v=−2xdx+(2x+2y)dy+c=−x+2xy+y+c,0022222f(z)=u+iv=(x+2xy−y)+i(−x+2xy+y+c)=(1−i)z+ic8.(1)1V)u=ρ2co

32、s2ϕ,122vρ=−ρuϕ=ρ3sin2ϕ,vϕ=ρuϕ=−ρ2cos2ϕ,221dv=sin2ϕdρ−cos2ϕdϕ=d(−sin2ϕ)ρ3ρ2ρ21v=−sin2ϕ+c,ρ211−i2ϕ1f(z)=(cos2ϕ−isin2ϕ)+