Q5P - Mathematical Methods in Physical Sciences

Question

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

$\mathit{l}\mathit{n}\mathbf{(}\mathbf{-}\sqrt{\mathbf{2}}\mathbf{ }\mathit{i}\mathbf{-}\sqrt{\mathbf{2}}\mathbf{)}$

Step-by-Step Solution

Answer

The form of the given equation $ln(-\sqrt{2} i-\sqrt{2})$ is,$\ln \left(2\right)+\left(\frac{5\pi }{4+2n\pi }4+2n\pi \right)i.$ .

 

Step 1: Given Information.

The given expression is, $ln(-\sqrt{2} i-\sqrt{2})$ .

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 is x the real part and y is the imaginary part.

Step 3: Convert in polar form.

Consider$z=-\sqrt{2}-i\sqrt{2.}$

 

Write the polar form of the number.

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

 

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

$$\begin{gathered} r=\left|-\sqrt{2}i-\sqrt{2}\right| \\ r=2 \\ \theta =\pi +\frac{\pi }{4} \\ \theta =\frac{5\pi }{4} \end{gathered}$$

 

Put the values in the polar form.

$z=2e^{\left(\frac{5\pi }{4}\right)}$

Step 4: Write in the form of .

Convert the polar form into the rectangular form.

$$\begin{gathered} w=\ln \left(r^{\left(\theta i\right)}\right) \\ w=\ln ( r)+\left(\frac{5\pi }{4}+2n\pi \right)i \\ w=\ln ( 2)+\left(\frac{5\pi }{4}+2n\pi \right)i \\ w=\ln ( r)+\left(\frac{5\pi }{4}+2n\pi \right)i where n=0,\pm 1,\pm 2,\pm 3,... \end{gathered}$$

Therefore, the form of the given equation $ln(-\sqrt{2} i-\sqrt{2})$ is, $\ln ( 2)+\left(\frac{5\pi }{4}+2n\pi \right)i$ .