Q3P - Mathematical Methods in Physical Sciences

Question

Evaluate each of the following in $\mathit{x}\mathbf{ }\mathbf{+}\mathbf{ }\mathit{i}\mathit{y}$ form, and compare with a computer solution.

$\mathit{l}\mathit{n}\mathbf{(}\mathit{i}\mathbf{+}\sqrt{\mathbf{3}}\mathbf{)}$

Step-by-Step Solution

Answer

The $x+iy$ form of the given equation $\ln (i+\sqrt{3})$ is $\ln \left(2\right)+\left(\frac{\mathrm{\pi }}{6}+2n\mathrm{\pi }\right)i$ .

Step 1: Given Information.

The given expression is $\ln (i+\sqrt{3})$ .

Step 2: Meaning of rectangular form.

Represent 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: Convert in polar form.

Consider.$z=i+\sqrt{3}$

 

Write the polar form of the number.

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

 

The angle is located in negative imaginary axis so the angle must be accordingly.

$$\begin{gathered} r=\left|i+\sqrt{3}\right| \\ r=2 \\ \theta =\frac{\mathrm{\pi }}{6} \end{gathered}$$

 

Put the values in the polar form.

$z=2e^{\left(\frac{\mathrm{\pi }}{6}i\right)}$

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

Convert the polar form into the rectangular form.

$$\begin{gathered} w =\ln \left(r{e}^{\left(\theta i\right)}\right) \\ w =\ln \left(r\right)+\left(\frac{\mathrm{\pi }}{6}+2n\mathrm{\pi }\right)i \\ w =\ln \left(2\right)+\left(\frac{\mathrm{\pi }}{6}+2n\mathrm{\pi }\right)i \\ w =\ln \left(2\right)+\left(\frac{\mathrm{\pi }}{6}+2n\mathrm{\pi }\right)i where n=0,\pm 1,\pm 2,\pm 3,....... \end{gathered}$$

Therefore, the $x+iy$ form of the given equation $\ln \left(i+\sqrt{3}\right)$ is $\ln \left(2\right)+\left(\frac{\mathrm{\pi }}{2}+2n\mathrm{\pi }\right)i$ .