\(
\def \l {\left}
\def \r {\right}
\def \f {\frac}
\def \b#1{\l(#1\r)}
\def \root [#1]#2{\sqrt[\leftroot{2}\uproot{2}\scriptstyle #1]{#2}}
\def \sroot [#1]#2{\sqrt[\large #1]{#2}}
\def \cbrt #1{\root[3]{#1}}
\def \scbrt #1{\sroot[3]{#1}}
\def \p {\phantom}
\def \stag#1{\quad (#1)}
\def \box#1{\boxed{\displaystyle{#1}}}
\def \und#1{\style{text-decoration: underline}{#1}}
\def \link#1#2{\und{\href{#1}{#2}}}
\def \tlink#1#2{\link{#1}{\text{#2}}}
\DeclareMathOperator{\acoth}{acoth}
\)
Square Root of Any Complex Number
Pub: Sep. 14, 2014 | Wri: a few years ago
\[
\sqrt{x \pm yi} \\[6pt]
= \sqrt{x \pm \sqrt{-y^2}} \\[12pt]
\begin{split}
a, b &= \f{x \pm \sqrt{x^2 - (-y^2)}}{2} \qquad\qquad
\text{(Using }
\tlink{denest-sqrt.html}{Denesting $\sqrt{x\pm\sqrt{y}}$}
\text{)} \\[3pt]
&= \f{x \pm \sqrt{x^2 + y^2}}{2}
\end{split} \\[30pt]
\begin{split}
\sqrt{x \pm yi}
&= \sqrt{\f{x + \sqrt{x^2 + y^2}}{2}} \pm
\sqrt{\f{x - \sqrt{x^2 + y^2}}{2}} \\[6pt]
&= \sqrt{\f{x + \sqrt{x^2 + y^2}}{2}} \pm \sqrt{\f{-x + \sqrt{x^2 + y^2}}{2}}i
\end{split}
\]
Square Root of $\pm i$
\[
\sqrt{\pm i} \\[6pt]
= \sqrt{0 \pm 1i} \\[6pt]
= \sqrt{\f{0 + \sqrt{0 + 1}}{2}} \pm \sqrt{\f{-0 + \sqrt{0 + 1}}{2}}i \\[12pt]
= \sqrt{\f{1}{2}} \pm \sqrt{\f{1}{2}}i \\[12pt]
= \f{1}{\sqrt{2}} \pm \f{1}{\sqrt{2}}i \\[12pt]
= \f{\sqrt{2}}{2} \pm \f{\sqrt{2}}{2}i \\[12pt]
\]