复数题Z=(1+i)/2^0.5

问题描述:

复数题Z=(1+i)/2^0.5
已知Z=(1+i)/2^0.5
则1+Z^50+Z^100的值是?

Z=(1+i)/2^0.5
则Z=2^0.5(cosπ/4+isinπ/4)/2^0.5=cosπ/4+isinπ/4
根据棣莫佛定理:
z^50=cos50π/4+isin50π/4=i
z^100=cos100π/4+isin100π/4=-1

1+Z^50+Z^100=1+i-1=i