Q1P - Mathematical Methods in Physical Sciences

Question

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

$\mathit{l}\mathit{n}\mathbf{(}\mathbf{-}\mathit{e}\mathbf{)}$

Step-by-Step Solution

Answer

The x+iy form of the given equation $(-e)$ is i $(1+\pi i)$.

Step 1: Given Information.

The given expression is $(-\mathrm{e})$.

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 $z=-e$.

 

Write the polar form of the number.

$z=r{e}^{\left(i\theta \right)}$ 

 

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

$\begin{array}{l}r=\left|-e\right|\\ r=e\\ \theta =\mathrm{\pi }\end{array}$ 

 

 

Put the values in the polar form.

$z=e.e^{\left(\pi i\right)}$ 

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

Convert the polar form into the rectangular form.

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

 

Therefore, the x+iy form of the given equation is, $\ln (-e)$  is  $\left(1+\pi i\right)$.