Q16P - Mathematical Methods in Physical Sciences

Question

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

${\left(\frac{1+i\sqrt{3}}{2}\right)}^{\mathbf{\text{'}}}$

Step-by-Step Solution

Answer

The value of $\left(\frac{1+i\sqrt{3}}{2}\right)$is Where $n=0,1,2,3..$.

Step 1: Given Information.

The given expression is ${\left(\frac{1+i\sqrt{3}}{2}\right)}^{\text{'}}$.

Step 2: Definition of Complex Number.

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

Step 3: Find the value of ( 1 + i 3 2 ) ' .

Let the complex numbers as:

$$\begin{gathered} u={\left(\frac{1+i\sqrt{3}}{2}\right)}^{\text{'}} \\ z=\frac{1+i\sqrt{3}}{2} \end{gathered}$$

 

Find the modulus of the complex number.

 $$\begin{gathered} r=\left|z\right| \\ =1 \end{gathered}$$.

Find the argument of the complex number.

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

 

Hence the value of ${\left(\frac{1+i\sqrt{3}}{2}\right)}^{\text{'}}$is $e^{\left[\left(-\frac{\mathrm{\pi }}{3}\underline{+}2\mathrm{n\pi }\right)\right]}$Where $n=0,1,2,3,..$