Question

已知z属于复数,z的模为,z不等于正负1,求证：z-1/z+1是纯虚数

Answer 1

设z=a+bi,且a²+b²=1

(z-1)/(z+1)

=(a+bi-1)/(a+bi+1)

=(a-1+bi)/(a+1+bi)

=(a-1+bi)(a+1-bi)/(a+1+bi)(a+1-bi)

=[a+(bi-1)][(a-(bi-1)]/[(a+1)²+b²]

=[a²-(bi-1)²]/(a²+2a+1+b²)

=2bi/(2a+2)

=bi/(a+1)

即z-1/z+1是纯虚数