设f(x)=x^100+x^50+1,求f[(-1-i)/根号2)]之值

设f(x)=x^100+x^50+1,求f[(-1-i)/根号2)]之值
设f(x)=x^100+x^50+1,求f[(-1-i)/根号2)]之值

设f(x)=x^100+x^50+1,求f[(-1-i)/根号2)]之值
x=(-1-i)/2
=-1/2-i/2
=(√2/2)(-√2/2-√2i/2)
=(√2/2)(cos5π/4+isin5π/4)
所以x^100=(√2/2)^100*[cos(100*5π/4)+isin(100*5π/4)]
=(1/2)^50*(cos125π+isin125π)
=(1/2)^50*(-1)
=-(1/2)^50
同理
x^50=(1/2)^25*(cos125π/2+isin125π/2)
=(1/2)^25*i
所以原式=-(1/2)^50+(1/2)^25*i+1