Q2P - 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.

$\mathbf{ln}\mathbf{(}\mathbf{-}\mathit{i}\mathbf{)}$

Step-by-Step Solution

Answer

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

Step 1: Given Information.

The given expression is $\ln (-i)$ .

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, $z=-i$.

 

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.

$$\begin{gathered} r=\left|-i\right| \\ r=1 \\ \theta =\frac{3\mathrm{\pi }}{2} \end{gathered}$$

 

Put the values in the polar form.

$z=e^{\left(\frac{3\mathrm{\pi }}{2}i\right)}$

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

Convert the polar form into the rectangular form.

$$\begin{gathered} z=r{e}^{\left(\theta i\right)} \\ \ln z =\ln \left(r\right) +\ln e^{\left(\theta i\right)} \\ \ln (-i)=\ln \left(1\right)+\frac{3\mathrm{\pi }}{2}i \ln e \end{gathered}$$

 

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