Question

Find one or more values of each of the following complex expressions and compare with a computer solution.

$\left(-i\right)\mathbf{\text{'}}$

Short answer

The value of $\left(-i\right)\text{'}$is,$e^{-\left(\frac{\mathrm{\pi }}{2}\pm 2n\mathrm{\pi }\right)}$ .

Step-by-step solution

Given Information

The given expression is,$\left(-i\right)\text{'}$ .

Definition of Complex Numbers

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

 Use the logarithmic functions to obtain the result

Consider,$w=\left(-i\right)\text{'}$ .

Write the number in the form, $w=e^{iIn\left(-i\right)}$. …. (1)

Use logarithmic functions to find the value of In$\left(-i\right)$ .

$In\left(-i\right)=In\left(r\right)+i\left(\theta \pm 2n\mathrm{\pi }\right)n=0,1,2,3,.....$

$$\begin{gathered} =In\left(1\right)+i\left(\frac{\mathrm{\pi }}{2}\pm 2n\mathrm{\pi }\right) \\ =i\left(\frac{\mathrm{\pi }}{2}\pm 2n\mathrm{\pi }\right)........\left(2\right) \end{gathered}$$

Put the value of In $\left(-i\right)$in equation (1).

$$\begin{gathered} w=e^{\left[i.i\left(\frac{\mathrm{\pi }}{2}\pm 2n\mathrm{\pi }\right)\right]} \\ w=e^{\left[-\left(\frac{\mathrm{\pi }}{2}\pm 2n\mathrm{\pi }\right)\right]}n=0,1,2,3,....... \end{gathered}$$

Hence the value of ${\left(-i\right)}^i$is, $e^{-\left(\frac{\mathrm{\pi }}{2}\pm 2n\mathrm{\pi }\right)}$.