# some values characterising i

• $\displaystyle i^{\,2}\;=\;-1$

• $|i|\;=\;1$

• $\displaystyle\arg{i}\;=\;2n\pi\!+\!\frac{\pi}{2}$  ($n\in\mathbb{Z}$)

• $\displaystyle\bar{i}\;=\;-i$

• $\displaystyle\frac{1}{i}\;=\;-i$

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

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

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

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

• $\displaystyle\sin{i}\;=\;\frac{i}{2}\left(e-\frac{1}{e}\right)\;\approx\;1.175% 20\,i$

• $\displaystyle\cosh{i}\;=\;\cos 1\;\approx\;0.54030$

• $\displaystyle\sinh{i}\;=\;i\,\sin 1\;\approx\;0.84147\,i$

• $\displaystyle e^{i}\;=\;\cos 1+i\,\sin 1$

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

Title some values characterising i SomeValuesCharacterisingI 2013-03-22 18:31:29 2013-03-22 18:31:29 pahio (2872) pahio (2872) 7 pahio (2872) Result msc 12D99 GeneralPower ComplexSineAndCosine ComplexLogarithm