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We know \[i = \cos \dfrac{\pi }{2} + i\sin \dfrac{\pi }{2} = {e^{i\dfrac{\pi }{2}}}\] So the value of \[{i^i}\]will be

\[\begin{gathered}

\therefore {i^i} = {\left( {{e^{i\dfrac{\pi }{2}}}} \right)^i} \\

= {e^{{i^2}\dfrac{\pi }{2}}} \\

= {e^{ - \pi /2}} \approx 0.20787957635076193 \\

\end{gathered} \]

General solution:

\[\begin{gathered}

i = {e^{i(2n\pi + \dfrac{\pi }{2})}} \\

= {e^{i(4n + 1)\dfrac{\pi }{2}}} \\

\end{gathered} \]

\[\begin{gathered}

\therefore {i^i} = {\left( {{e^{i\dfrac{{(4n + 1)\pi }}{2}}}} \right)^i} \\

= {e^{{i^2}\dfrac{{(4n + 1)\pi }}{2}}} \\

= {e^{ - \dfrac{{(4n + 1)\pi }}{2}}} \\

\end{gathered} \]

Where, n is any integer i.e. n=0, ±1, ±2, ±3,… n = 0, ±1, ±2, ±3,…

Now, substituting the integral values of n, one should get infinitely many solutions as follows

\[{i^i} = \ldots {e^{ - \dfrac{{9\pi }}{2}}},{e^{ - \dfrac{{5\pi }}{2}}},{e^{ - \dfrac{\pi }{2}}},{e^{\dfrac{{3\pi }}{2}}},{e^{\dfrac{{7\pi }}{2}}}...\]

where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number, and so some authors refer to the more general complex version as Euler's formula.

Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".

When , Euler's formula evaluates to , which is known as Euler's identity.