解法一:
|z|=1 则|z|^2=1
u=z^2-(i-1)
|u|的几何意义就是单位圆(圆心为原点O 半径是1)上的点M与点P(1,-1)的距离
显然连接MO
设它与圆交点依次是A B
则│AP│=√2-1≤│MP│≤│BP│=√2+1
解法二:
|z|=1,故设z=cost+isint
u=z^2-i+1
=(cost+isint)^2+(1-i)
=(cos2t+isin2t)+(1-i)
=(cos2t+1)+i(sin2t-1)
|u|=√[(cos2t+1)^2+(sin2t-1)^2]
=√(1+1+1+2cos2t-2sin2t)
=√[3+2(cos2t-sin2t)]
=√[3+2√2cos(2t+pi/4)]
-1≤cos(2t+pi/4)≤1
3-2√2=