Q18P - Mathematical Methods in Physical Sciences

Question

For each of the following numbers, first visualize where it is in the complex plane. With a little practice you can quickly find  $\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{r}\mathbf{,}\mathit{\theta }$ in your head for these simple problems. Then plot the number and label it in five ways as in Figure 3.3. Also plot the complex conjugate of the number.

$\mathbf{3}{\mathit{e}}^{{\mathbf{i}}_{\mathbf{2}}^{\mathbf{x}}}$ .

Step-by-Step Solution

Answer

 

The required values are mentioned below:

 $x=0,y=3, r=3,\theta =\frac{\pi }{2}$

 

The graph of the number and its conjugate is shown below:

 

Step 1: Given Information

The complex number is $3e^{i_2^x}$.

Step 2: Definition of the Complex number.

A complex number is a combination of real and imaginary numbers.$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}{}_{\mathbf{z}}\mathit{b}$   Where a and b are real numbers and z is the complex number.

Step 3: Find the value

The formula is mentioned below:

 $$\begin{gathered} x+iy=r\left(\cos \theta +i\sin \theta \right) \\ =r{e}^{i\theta } \end{gathered}$$

 $3e^{i_2^x}=3\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)$

 

The formulas for x and y are given below:

$$\begin{gathered} x=r\cos \left(\theta \right) \\ y=r\sin \left(\theta \right) \end{gathered}$$

 

Find x and y:

$$\begin{gathered} x=3\cos \left(\frac{\pi }{2}\right) \\ =0 \\ y=3\sin \left(\frac{\pi }{2}\right) \\ =3 \end{gathered}$$

  

Find the value of $\theta$ .

Compare with the value of x:

$$\begin{gathered} x=3\cos \left(\frac{\pi }{2}\right) \\ r=3 \end{gathered}$$

 

Find the value of $\theta$ .

Compare with the value of x:

$$\begin{gathered} x=3\cos \left(\frac{\pi }{2}\right) \\ \theta =\frac{\pi }{2} \end{gathered}$$

  

The value of the number becomes as follows:

$3\cos \left(\frac{\pi }{2}\right)+i\sin \left(\frac{\pi }{2}\right)=3e^{i\left(x/2\right)}$

 

 The graph of the complex number (blue) and its conjugate (red)  is shown below:

 

 

 

The required values are mentioned below:

$x=0,y=3.r=3,\theta =\frac{\pi }{2}$