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Question Number 132632 by mohammad17 last updated on 15/Feb/21

$$Solvetheequation{z}^3=z_{°}$$$$z=x+iy,z_{°}=(x_{°}+{iy}_{°})$$

Answered by TheSupreme last updated on 15/Feb/21

$$x_0+{iy}_0=\rho e^{i\theta }\rho =\sqrt{x_0^2+y_0^2}tan\left(\theta \right)=\frac{y_0}{x_0}$$$$wehavetocorrectphase$$$$\theta =atan\left(\frac{x_0}{y_0}\right)+\pi sgn\left(x_0\right)$$$$z=z_0^{\frac{1}{3}}={\rho }^{\frac{1}{3}}{e}^{i\frac{\theta +2n\pi }{3}}$$$$z_n={(x_0^2+y_0^2)}^{\frac{1}{6}}{e}^{i\frac{atan\left(\frac{x_0}{y_0}\right)+\pi (2n+sgn\left(x_0\right))}{3}}i=0,1,2$$$$$$

Commented by mohammad17 last updated on 15/Feb/21

$$sircanyougivemedefintionofsgnplease$$

Commented by Ar Brandon last updated on 15/Feb/21

$$\mathrm{sgn}\left(\mathrm{x}\right)=\mathrm{sign}\mathrm{of}\mathrm{x}$$

Commented by mr W last updated on 15/Feb/21

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