Question

已知复数Z满足：|Z|=1且Z不等于正负i,求证：Z/1+Z²是实数.

Answer 1

设Z=a+bi

|Z|=1且Z不等于正负i,a≠0,b≠±1

a²+b²=1

a²=1-b²

Z/(1+Z²)

=(a+bi)/[1+(a+bi)²]

=(a+bi)/(a²+1-b²+2abi)

=(a+bi)/(a²+a²+2abi)

=(a+bi)/(2a²+2abi)

=(a+bi)/[2a(a+bi)]

=1/2a是实数