<s>[INST] Solving roots of complex number Given $z= \dfrac{i-1}{2}$. Evaluate $z^{1/2}$ and show the roots.
I got $z^{1/2}=\dfrac{1}{2^{1/4}}\left(\cos\dfrac{3\pi}{8} + i\sin\dfrac{3\pi}{8}\right)$
First of all is my $z^{1/2}$ correct? And secondly how to show the roots??
I would be glad and appreciated if someone could help me out.
 [/INST] For computing $z^{1/2}=\sqrt{x+yi}$ put $$\sqrt{x+yi}=a+bi$$ and so $(\sqrt{x+yi})^2=(a+bi)^2$. If you do the latter identity, you will find $$(1): a^2-b^2=x\\(2):2ab=y\\(3):(a^2+b^2)^2=\sqrt{x^2+y^2}$$ Adding (1) and (3) gives you $$2a^2=\sqrt{x^2+y^2}+x$$ and subtracting (1) of (3) gives you $$2b^2=\sqrt{x^2+y^2}-x$$ Now, the rest is easy.
</s>