Question

Evaluate each of the following in x+ iyform, and compare with a computer solution.

${\mathit{i}}^{\mathit{2}\mathit{i}\mathit{/}\mathit{\pi }}$

Short answer

The value of $i^{\frac{2i}{\mathrm{\pi }}}$is $e^{\left[-1\overline{+}2n\mathrm{\pi }\right]}$Where $n=0,1,2,3,....$.

Step-by-step solution

Given Information.

The given number is $i^{\frac{2i}{\mathrm{\pi }}}$.

Definition of Complex Numberand Hyperbolic functions.

The numbers that are presented in the form of $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$where, $\mathbf{\text{'}}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{\text{'}}$are real numbers and role="math" localid="1658812700788" $\mathit{\text{'}}\mathit{i}\mathit{\text{'}}$ is an imaginary number, those numbers are referred to as called Complex numbers.

Find the value of i2iπ.

$$\begin{gathered} i^{\frac{2i}{\mathrm{\pi }}}=e^{\left[\mathrm{In}\left(i^{\frac{2i}{\mathrm{\pi }}}\right)\right]} \\ =e^{\left[\frac{2i}{\mathrm{\pi }}\left(In\left(1\right)+i\right)\left(\frac{\mathrm{\pi }}{2}\underline{+}2n\mathrm{\pi }\right)\right]} \\ =e^{\left[\frac{2i}{\mathrm{\pi }}\left(i\left(\frac{\mathrm{\pi }}{2}\underline{+}2n\mathrm{\pi }\right)\right)\right]} \\ =e^{\left[-1\overline{+}2n\mathrm{\pi }\right]} \end{gathered}$$

Where$n=0,1,2,3,4,....$

Hence, the solution is in the form of $e^{\left[-1\overline{+}2n\mathrm{\pi }\right]}$Where$n=0,1,2,3,...$