Q6P - Mathematical Methods in Physical Sciences

Question

Question: Evaluate each of the following in x+y  form, and compare with a computer solution.\

$\ln \left(\frac{1-i}{\sqrt{2}}\right)$

Step-by-Step Solution

Answer

The x+y form of the given equation $\ln \left(\frac{1-i}{\sqrt{2}}\right)$is $\left(\frac{7\pi }{4}+2n\pi \right)i.$

Step 1: Given Information.

The given expression is, $\ln \left(\frac{1-i}{\sqrt{2}}\right)$

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.

Consider $z=\left(\frac{1-i}{\sqrt{2}}\right)$

 

Write the polar form of the number.

$z=r{e}^{\left(i\theta \right)}$

 

The angle is located in fourth quadrant so the angle must be accordingly.

$$\begin{gathered} r=\left|\frac{1-i}{\sqrt{2}}\right| \\ r=1 \\ \theta =2\pi -\frac{\pi }{4} \\ \theta =\frac{7\pi }{4} \end{gathered}$$

 

Put the values in the polar form.

$z=e^{\left(\frac{7\pi }{4}i\right)}$

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

Convert the polar form into the rectangular form.

$$\begin{gathered} w=\ln \left(r{e}^{\left(\theta i\right)}\right) \\ w=\ln \left(r\right)+\left(\frac{7\pi }{4}+2n\pi \right)i \\ w=\ln \left(1\right)+\left(\frac{7\pi }{4}+2n\pi \right)i \\ w=\ln \left(1\right)+\left(\frac{7\pi }{4}+2n\pi \right)i where n=0,\pm 1,\pm 2,\pm 3,\dots \end{gathered}$$

Therefore, the form of the given equation $\ln \left(\frac{1-i}{\sqrt{2}}\right)$ is,$\left(\frac{7\pi }{4}+2n\pi \right)i$ .