Q17P - 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  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.

 $\sqrt{\mathbf{2}}{\mathit{e}}^{\mathbf{-}\mathbf{i}\frac{\mathbf{\pi }}{\mathbf{4}}}$. 

Step-by-Step Solution

Answer

The required values are mentioned below:

 $x=1,y=-1,r=\sqrt{2},\theta =-\frac{\pi }{4}$

 

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

 

Step 1: Given Information

The complex number is $\sqrt{2}{e}^{-i\frac{\pi }{4}}$.

Step 2: Definition of the Complex number.

Real numbers can be written as complex numbers with the help of iota (i).

$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}\mathit{\tau }\mathit{b}$  where a and b are real numbers.

Step 3: Find the value.

The formula is mentioned below:

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

 

$\sqrt{2}{e}^{-i\frac{\pi }{4}}\sqrt{2}cos\frac{\pi }{4}-isin\frac{\pi }{4}$

The formulas for x and y are given below:

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

 

Find x and y:

 $$\begin{gathered} x=\sqrt{2}\cos \left(-\frac{\pi }{4}\right) \\ =1 \\ y=\sqrt{2}\sin \left(-\frac{\pi }{4}\right) \\ =-1 \end{gathered}$$

 

Find the value of r :

Compare with the value of x.

 $\begin{array}{l}x=\sqrt{2}cos\left(-\frac{\mathrm{\pi }}{4}\right)\\ r=\sqrt{2}\end{array}$

 

Find the value of $\theta$.

Compare with the value of x.

 $$\begin{gathered} x=\sqrt{2 }\cos \left(-\frac{\pi }{4}\right) \\ \theta =-\frac{\pi }{4} \end{gathered}$$

 

The value of the number becomes as follows:

 $\sqrt{2}\left(\cos \left(-\frac{\pi }{4}\right)-i\sin \left(\frac{\pi }{4}\right)\right)=\sqrt{2}{e}^{-i\left(\pi /4\right)}$

 

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

 

The required values are mentioned below:

$x=1,y=-1,r=\sqrt{2},\theta =-\frac{\pi }{4}$