Q8P - Mathematical Methods in Physical Sciences

Question

Evaluate each of the following in $\mathbf{x}\mathbf{+}\mathbf{iy}$ form, and compare with a computer solution ${\mathbf{i}}^{\mathbf{2}\mathbf{/}\mathbf{3}}$.

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{The} \mathrm{x}+\mathrm{iy} \mathrm{form} \mathrm{of} \mathrm{the} \mathrm{given} \mathrm{equation}, {\mathrm{i}}^{2/3}. \\ {\mathrm{w}}_1=\frac{1}{2}+\mathrm{i}\frac{\sqrt{3}}{2} \\ {\mathrm{w}}_2=\frac{1}{2}-\mathrm{i}\frac{\sqrt{3}}{2} \\ {\mathrm{w}}_3=-1 \end{gathered}$$

Step 1: Given Information.

The given expression is $i^{2/3}$.

Step 2: Meaning of rectangular form.

Represent the complex number in rectangular form means writing the given complex number in the form of x+iy in which x is the real part and y is the imaginary part.

Step 3: Convert in polar form.

$$\begin{gathered} Consider, w=i^{2/3} \\ Write the given number in polar form. \\ w=e^{\left[In{\left(i\right)}^{2/3}\right]} \\ Use the \log arithm property. \\ w=e^{\left[\frac{2}{3}In\left(i\right)\right]} \\ w=e^{\left(\frac{2}{3}\left(In r+\left(\theta +2n\pi \right)i\right)\right)} \\ w=e^{\left(\frac{2}{3}\left(\frac{\pi }{2}+2n\pi \right)i\right)} \\ w=e^{\left(\left(\frac{\pi }{3}+\frac{4n\pi }{3}\right)i\right)} \\ Where n=0,\pm 1,\pm 2,\pm 3,..... \end{gathered}$$

Step 4: Write in the form of x + iy .

Use the definition of so In and $\theta =\frac{\pi }{2}$because the number is pure imaginary.

It’s the third root so it has three values.

 

Put the values.

Put n=0

$$\begin{gathered} w_1=e^{\left(\pi /3\right)} \\ {w}_1=\frac{1}{2}+i\frac{\sqrt{3}}{2} \end{gathered}$$

 

Put n=1

$$\begin{gathered} w_2=e^{\left(5\pi /3\right)} \\ {w}_2=\frac{1}{2}-i\frac{\sqrt{3}}{2} \end{gathered}$$

 

Put n=-1

$$\begin{gathered} w_3=e^{\left(-\pi \right)} \\ {w}_3=-1 \end{gathered}$$

 

Therefore, x+iy form of the given equation $i^{2/3}$ .

$$\begin{gathered} w_1=\frac{1}{2}+i\frac{\sqrt{3}}{2} \\ {w}_2=\frac{1}{2}-i\frac{\sqrt{3}}{2} \\ {w}_3=-1 \end{gathered}$$