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what is the value of \[{i^i}\], where \[i = \sqrt { - 1} \]

Last updated date: 13th Jun 2024
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Hint: express \[i\] in Euler form. The Euler formula is the way to express complex quantities into trigonometric ratios, In particular sin and cos ratios. If we are going into trigonometry then $\theta$ will come into play.

Complete step by step solution: Given \[{i^i}\], where \[i = \sqrt { - 1} \]

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
  \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:
  i = {e^{i(2n\pi + \dfrac{\pi }{2})}} \\
   = {e^{i(4n + 1)\dfrac{\pi }{2}}} \\
\end{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
\[{i^i} = \ldots {e^{ - \dfrac{{9\pi }}{2}}},{e^{ - \dfrac{{5\pi }}{2}}},{e^{ - \dfrac{\pi }{2}}},{e^{\dfrac{{3\pi }}{2}}},{e^{\dfrac{{7\pi }}{2}}}...\]

Additional solution: Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number x:

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.

Note: \[{i^i}\] has an infinite number of solutions. All the solutions contain positive value. Exponentiating this equation yields Euler's formula. Note that the logarithmic statement is not universally correct for complex numbers, since a complex logarithm can have infinitely many values, differing by multiples of $2i\pi$.