z^3+i=0的全部解.

Question

z^3+i=0的全部解.

Answer 1

z³+i=0的全部解
z³=-i=cos(3π/2)+isin(3π/2)
z=cos[(3π/2+2kπ)/3]+isin[(3π/2+2kπ)/3]
当k=0时得z₀=cos(π/2)+isin(π/2)=i
当k=1时z₁=cos[(3π/2+2π)/3]+isin[(3π/2+2π)/3]
=cos(π/2+2π/3)+isin(π/2+2π/3)=-sin(2π/3)+icos(2π/3)
=-sin(π-π/3)+icos(π-π/3)=-sin(π/3)-icos(π/3)
=-(√3)/2-(1/2)i
当k=2时z₂=cos[(3π/2+4π)/3]+isin[(3π/2+4π)/3]
=cos[π/2+(4/3)π]+isin[π/2+(4π/3)]
=-sin[(4/3)π]+icos[(4π/3)]
=-sin(π+π/3)+icos(π+π/3)
=sin(π/3)-icos(π/3)
=(√3)/2-(1/2)i