Explicație pas cu pas:
[tex]z^{2} = i \overline{z}[/tex]
[tex]z = a + b i[/tex]
[tex]\overline{z} = a - bi[/tex]
=>
[tex] {(a + b i)}^{2} = i(a - b i) [/tex]
[tex]( {a}^{2} - {b}^{2} ) + 2ab \ i = b + a \ i[/tex]
=>
[tex]\begin{cases} {a}^{2} - {b}^{2} = b\\ 2ab = a\end{cases}[/tex]
dacă:
[tex]a = 0 \implies b(b + 1) = 0[/tex]
[tex]\bf \implies \begin{cases} a = 0; b = -1\\a = 0; b = 0 \end{cases} [/tex]
dacă:
[tex]a \not= 0 \implies 2b = 1 \implies b = \frac{1}{2} \\ [/tex]
[tex]{a}^{2} = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \\ [/tex]
[tex]\bf \implies \begin{cases} a = - \frac{ \sqrt{3} }{2} ; b = \frac{1}{2} \\a = \frac{ \sqrt{3} }{2} ; b = \frac{1}{2} \end{cases} [/tex]
