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