$\def \l {\left} \def \r {\right} \def \f {\frac} \def \b#1{\l(#1\r)} \def \root [#1]#2{\sqrt[\leftroot{2}\uproot{2}\scriptstyle #1]{#2}} \def \sroot [#1]#2{\sqrt[\large #1]{#2}} \def \cbrt #1{\root[3]{#1}} \def \scbrt #1{\sroot[3]{#1}} \def \p {\phantom} \def \stag#1{\quad (#1)} \def \box#1{\boxed{\displaystyle{#1}}} \def \und#1{\style{text-decoration: underline}{#1}} \def \link#1#2{\und{\href{#1}{#2}}} \def \tlink#1#2{\link{#1}{\text{#2}}} \DeclareMathOperator{\acoth}{acoth}$

# 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 $\pm i$

$\sqrt{\pm i} \\[6pt] = \sqrt{0 \pm 1i} \\[6pt] = \sqrt{\f{0 + \sqrt{0 + 1}}{2}} \pm \sqrt{\f{-0 + \sqrt{0 + 1}}{2}}i \\[12pt] = \sqrt{\f{1}{2}} \pm \sqrt{\f{1}{2}}i \\[12pt] = \f{1}{\sqrt{2}} \pm \f{1}{\sqrt{2}}i \\[12pt] = \f{\sqrt{2}}{2} \pm \f{\sqrt{2}}{2}i \\[12pt]$