Q7MP - Mathematical Methods in Physical Sciences

Question

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

 $\left(-i\right)\mathbf{\text{'}}$

Step-by-Step Solution

Answer

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

Step 1: Given Information

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

Step 2: 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.

Step 3: Use the logarithmic functions to obtain the result

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

Write the number in the form, $w=e^{i In\left(-i\right)}$ .                                                                  …. (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, $e^{-\left(\frac{\mathrm{\pi }}{2}\pm 2n\mathrm{\pi }\right)}$.