Q17P - Mathematical Methods in Physical Sciences

Question

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

${\left(i-1\right)}^{\mathbf{l}\mathbf{+}\mathbf{1}}$

Step-by-Step Solution

Answer

The value of  $\left(\mathrm{i}-1\right)\mathrm{l}+1 \mathrm{is} {\mathrm{e}}^{\left[-\left(\frac{3\mathrm{\pi }}{4}\pm 2\mathrm{n\pi }\right)-\mathrm{In}\left(\sqrt{2}\right)\right]}.\left[-0.9+\mathrm{i}0.42\right] \mathrm{Where} \mathrm{n} = 0,1,2,3,.....$

Step 1: Given Information.

 The given number is ${\left(\mathrm{i}-1\right)}^{\mathrm{i}+1}$ .

Step 2: Definition of Complex Number.

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

Step 3: Find the value of ( i - 1 ) i + 1 .

Let the complex number as:

$$\begin{gathered} u={\left(i-1\right)}^{i-1} \\ z=\left(i-1\right) \end{gathered}$$

 

Find the modulusof the complex number.

 $$\begin{gathered} r=\left|z\right| \\ =\left|i-1\right| \\ =\sqrt{2} \end{gathered}$$

 

Find the argument of the complex number. 

$$\begin{gathered} \theta =\mathrm{\pi }-\left|\arctan 1\right| \\ =\frac{3\mathrm{\pi }}{4} \end{gathered}$$

$$\begin{gathered} u=e^{\left(In{\left(i-1\right)}^{i-1}\right)} \\ =e\left[\left(i+1\right)In\left(i-1\right)\right] \\ =e\left[\left(i+1\right)\left[In\left(r\right)+i\left(\theta \pm 2n\mathrm{\pi }\right)\right]\right]......\left(1\right) \\ \mathit{n}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.} \end{gathered}$$

Put the value of r and $\theta$ in equation (1).

$$\begin{gathered} u=e^{\left[\left(i-1\right)\left[In\left(\sqrt{2}\right)+i\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)\right]\right]} \\ =e^{\left[In\left(\sqrt{2}\right)+i\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)+In\left(\sqrt{2}\right)+i\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)\right]} \\ =e^{\left[In\left(\sqrt{2}\right)+i\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)\left(\sqrt{2}\right)+i\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)\right]}......\left(2\right) \end{gathered}$$

Separately solve the expression $e^{\left[iIn\left(\sqrt{2}\right)+i\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)\right]}$.

$e^{\left[iIn\left(\sqrt{2}\right)+i\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)\right]}$ $$\begin{gathered} =\left[\cos \left(In\left(\sqrt{2}\right)+\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)+i \sin \left(In\left(\sqrt{2}\right)+\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)\right] \\ =\left[\cos \left(In\left(\sqrt{2}\right)+\frac{3\mathrm{\pi }}{4}\right)+i \sin \left(In\left(\sqrt{2}\right)+\frac{3\mathrm{\pi }}{4}\right)\right] \\ =\left[-0.9+i0.42\right].....\left(3\right) \end{gathered}$$

Put the value of equation (3) in equation (2).

$$\begin{gathered} u=e^{\left[i In\left(\sqrt{2}\right)+i\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)\right]} e^{\left[-\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)+In\left(\sqrt{2}\right)\right]} \\ =e^{\left[-\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)+In\left(\sqrt{2}\right)\right]}.\left[-0.9+i0.42\right] \\ n=0,1,2,3..... \end{gathered}$$

 

Hence the value of is ${\left(i-1\right)}^{i-1}$ $=e^{\left[-\left(\frac{3\mathrm{\pi }}{4}\pm 2n\mathrm{\pi }\right)+In\left(\sqrt{2}\right)\right]}.\left[-0.9+i0.42\right] Where n=0,1,2,3.....$