The complex number is
z=-1+i
And we need z^(1/3)
The polar form of z is
z=|z|(costheta+isintheta)
where,
{(|z|=sqrt((-1)^2+(1)^2)),(costheta=-1/|z|),(sintheta=1/|z|):}
=>, {(|z|=sqrt2),(costheta=-1/sqrt2),(sintheta=1/sqrt2):}
=>, theta=3/4pi+2kpi
Therefore,
z=(sqrt2)^(1/3)(cos(3/4pi+2kpi)+isin(3/4pi+2kpi))
So, By De Moivre's theorem
z^(1/3)=(sqrt2)^(1/3)(cos(1/4pi+2/3kpi)+isin(1/4pi+2/3kpi))
When
k=0, =>, z_0=(sqrt2)^(1/3)(cos(1/4pi)+isin(1/4pi))
=(sqrt2)^(1/3)(1/sqrt2+i/sqrt2)
=2^(-1/3)+i2^(-1/3)
=0.7937+i0.7937
k=1, =>, z_1=(sqrt2)^(1/3)(cos(11/12pi)+isin(11/12pi))
=-1.084+i0.2905
k=2, =>, z_2=(sqrt2)^(1/3)(cos(19/12pi)+isin(19/12pi))
=0.2905-i1.084
The solution is {0.7937+i0.7937, -1.084+i0.2905, 0.2905-i1.084 }
