some values characterising i

• $i^{\mathrm{\hspace{0.17em}2}}=-1$

• $|i|=\mathrm{\hspace{0.33em}1}$

• $\arg i=\mathrm{\hspace{0.33em}2}n\pi +{\displaystyle \frac{\pi }{2}}$  ($n\in \mathbb{Z}$)

• $\overline{i}=-i$

• ${\displaystyle \frac{1}{i}}=-i$

• $\sqrt{i}=\pm {\displaystyle \frac{1+i}{\sqrt{2}}}$

• ${(-1)}^i=e^{(2n+1)\pi }$   ($n\in \mathbb{Z}$)

• $i^i=e^{2n\pi -\frac{\pi }{2}}$  ($n\in \mathbb{Z}$)

• $\cos i={\displaystyle \frac{1}{2}}\left(e+{\displaystyle \frac{1}{e}}\right)\approx \mathrm{\hspace{0.33em}1.54308}$

• $\sin i={\displaystyle \frac{i}{2}}\left(e-{\displaystyle \frac{1}{e}}\right)\approx \mathrm{\hspace{0.33em}1.17520}i$

• $\cosh i=\cos 1\approx \mathrm{\hspace{0.33em}0.54030}$

• $\sinh i=i\sin 1\approx \mathrm{\hspace{0.33em}0.84147}i$

• $e^i=\cos 1+i\sin 1$

• $\log i=\left(2n\pi +{\displaystyle \frac{\pi }{2}}\right)i$  ($n\in \mathbb{Z}$)