Question Number 195035 by mathlove last updated on 22/Jul/23
$$\sqrt{{i}}=? \\ $$
Answered by alephzero last updated on 22/Jul/23
$$\sqrt{{i}}\:=\:? \\ $$$$\left(\sqrt{{i}}\right)^{\mathrm{2}} \:=\:{i} \\ $$$${e}^{{i}\pi} \:=\:−\mathrm{1} \\ $$$$\sqrt{{e}^{{i}\pi} }\:=\:{e}^{{i}\frac{\pi}{\mathrm{2}}} \:=\:{i} \\ $$$$\sqrt{{e}^{{i}\frac{\pi}{\mathrm{2}}} }\:=\:{e}^{{i}\frac{\pi}{\mathrm{4}}} \:=\:\sqrt{{i}} \\ $$$${e}^{{ix}} \:=\:\mathrm{cos}\:{x}\:+\:{i}\:\mathrm{sin}\:{x} \\ $$$${e}^{{i}\frac{\pi}{\mathrm{4}}} \:=\:\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\:+\:{i}\:\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\Rightarrow\:\sqrt{{i}}\:=\:\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\:+\:{i}\:\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$
Commented by mathlove last updated on 23/Jul/23
$${thanks}\:{sir} \\ $$
Answered by Frix last updated on 22/Jul/23
$$\mathrm{i}=\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{2}}} \\ $$$$\sqrt{\mathrm{i}}=\mathrm{i}^{\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{4}}} \\ $$