Q5P - Mathematical Methods in Physical Sciences

Question

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

${\mathit{e}}^{\left(i\pi /4\right)\mathbf{+}\left(ln2\right)\mathbf{/}\mathbf{2}}$.

Step-by-Step Solution

Answer

The rectangular form of the given question is $e^{\left(\frac{i\pi }{4}\right)+\frac{\left(ln2\right)}{2}}=1+i$.

Step 1: Given Information.

The given expression is $e^{\left(i\pi /4\right)+\left(\ln 2\right)/2}$.

Step 2: Meaning of rectangular form.

Representing 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: Separate the exponential.

The given question is $e^{\left(i\pi /4\right)+\left(\ln 2\right)/2}$.

 

Break the exponential part in the given question.

$$$$\begin{gathered} e^{\left(\frac{i\pi }{4}\right)+\frac{\left(\ln 2\right)}{2}}=e^{\frac{i\pi }{4}}{e}^{\left(\frac{\ln 2}{2}\right)} \\ {e}^{\left(\frac{i\pi }{4}\right)+\frac{\left(\ln 2\right)}{2}}=e^{\frac{i\pi }{4}}{e}^{\left(\frac{1}{2}\ln 2\right)} \\ {e}^{\left(\frac{i\pi }{4}\right)+\frac{\left(\ln 2\right)}{2}}=e^{\frac{i\pi }{4}}{e}^{\left(\ln 2^{\frac{1}{2}}\right)} \\ {e}^{\left(\frac{i\pi }{4}\right)+\frac{\left(\ln 2\right)}{2}}=e^{\frac{i\pi }{4}}{2}^{\frac{1}{2}} \end{gathered}$$

Step 4: Convert it into rectangular form.

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

$$\begin{gathered} e^{\frac{i\pi }{4}}{2}^{\frac{1}{2}}=\left(\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}\right)2^{\frac{1}{2}} \\ =\frac{\left(1+i\right)}{\sqrt{2}}\sqrt{2} \\ =1+i \end{gathered}$$

 

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