Question

函数w=1/z,把z平面上x^2+(y-1)^2=4映射成w平面上怎样的曲线?

Answer 1

z=1/w=1/(u+iv)=(u-iv)/(u^2+v^2)

z=x+iy

x=u/(u^2+v^2),.u^2+v^2>0

y=-v/(u^2+v^2),

(u/(u^2+v^2))^2+(-v/(u^2+v^2)-1)^2=4

u^2+(u^2+v^2+v)^2=4(u^2+v^2)^2

u^2+v^2+2v(u^2+v^2)+(u^2+v^2)^2=4(u^2+v^2)^2

(1+2v)(u^2+v^2)=3(u^2+v^2)^2

1+2v=3(u^2+v^2)

u^2+(v-1/3)^2=4/9

是w平面上以（0,1/3）为圆心,2/3为半径的圆