Q13P - Mathematical Methods in Physical Sciences

Question

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

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

Step-by-Step Solution

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 1: Given Information.

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

Step 2: 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 $\mathit{\text{'}}\mathit{i}\mathit{\text{'}}$ is an imaginary number, those numbers are referred to as called Complex numbers.

Step 3: Find the value of i 2 i π .

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

 

Where $n=0,1,2,3,4,....$

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