Q3P - Mathematical Methods in Physical Sciences

Question

Find each of the following in rectangular form $\mathbf{x}\mathbf{+}\mathbf{iy}$ and check your results by computer. Remember to save time by doing as much as you can in your head.

${\mathbf{e}}^{\mathbf{-}\left(i\pi /4\right)\mathbf{+}\mathbf{ln}\mathbf{3}}$.

Step-by-Step Solution

Answer

The rectangular form of the given question is $e^{-\left(\frac{i\pi }{4}\right)+\ln 3}=\frac{3}{\sqrt{2}}\left(1-i\right)$.

Step 1: Given Information.

The given expression is $e^{-\left(i\pi /4\right)+\ln 3}..$

Step 2: Meaning of rectangular form.

Representing the complex number in rectangular form means writing the given complex number in the form of $\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$, in which x is the real part and y is the imaginary part.

Step 3: Separate the exponential.

The given question is $e^{-\left(i\pi /4\right)+\ln 3}$.

Break the exponential part in the given question.

$$\begin{gathered} e^{-\left(\frac{i\pi }{4}\right)+\ln 3}=e^{-\left(\frac{i\pi }{4}\right)}{e}^{\ln 3} \\ {e}^{-\left(\frac{i\pi }{4}\right)+\ln 3}=e^{-\frac{i\pi }{4}}\times 3 \\ {e}^{-\left(\frac{i\pi }{4}\right)+\ln 3}=3e^{\frac{-i\pi }{4}} \end{gathered}$$

Step 4: Convert it into rectangular form.

Use the complex formula $e^{-i\theta }=\cos \theta -i \sin \theta$to rewrite the above expression.

$$\begin{gathered} 3e^{\frac{-i\pi }{4}}=3\left[\cos \left(\frac{\pi }{4}\right)-i \sin \left(\frac{\pi }{4}\right)\right] \\ =3\left(\frac{1}{\sqrt{2}}-i\frac{1}{\sqrt{2}}\right) \\ =\frac{3}{\sqrt{2}}\left(1-i\right) \end{gathered}$$

 

Therefore, the rectangular form of   $e^{-\left(i\pi /4\right)+\ln 3}$ is $\frac{3}{\sqrt{2}}\left(1-i\right)$.