Q24P - Mathematical Methods in Physical Sciences

Question

Find each of the following in rectangular $\mathit{a}\mathbf{+}{}_{\mathbf{z}}\mathit{b}$  form if  $\mathit{z}\mathbf{=}\mathbf{2}\mathbf{-}{}_{\mathbf{t}}\mathbf{3}$ ; if $\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}{}_{\mathbf{z}}\mathit{y}$ .

 $\frac{\mathbf{z}}{\mathbf{z}}$.

Step-by-Step Solution

Answer

The value for $\frac{z}{z}$  is given below.

$$\begin{gathered} \frac{z}{z}=\frac{-5-12i}{13} \\ \frac{z}{\overline{z}}=\frac{x^2+2xyi-y^2}{x^2+y^2} \end{gathered}$$

 

Step 1: Given Information

The complex number is $\frac{z}{\overline{z}}$.

Step 2: Definition of the Complex number.

Complex number z can be written in the form of the real and imaginary numbers as:

$\mathit{z}\mathbf{=}\mathit{a}\mathbf{+}{}_{\mathbf{z}}\mathit{b}$

 

ere z is the complex number, and a and b are real numbers.

Step 3: Find the value.

$\frac{z}{\overline{z}}$The complex number is $\frac{1=z}{1-z}$ .

 

Find the values for $2-3i$ .

$$\begin{gathered} \frac{z}{\overline{z}}=\frac{2-3i}{2+3i} \\ \frac{2-3i}{2+3i}\times \frac{2-3i}{2-3i}=\frac{\left(2-3i\right)}{4-{\left(3i\right)}^2} \\ =\frac{4-12i-9}{4=9} \\ =\frac{-5-12i}{13} \end{gathered}$$

 

Find the values for $x+iy$ .

$$\begin{gathered} \frac{z}{\overline{z}}=\frac{x+yi}{z=x-yi} \\ \frac{x=yi}{x-yi}\times \frac{x+yi}{x+yi}=\frac{{\left(x+yi\right)}^2}{x^2-{\left(yi\right)}^2} \\ =\frac{x^2+2xyi-y^2}{x^2+y^2} \end{gathered}$$

The value for  $\frac{z}{\overline{z}}$ is given below:

$$\begin{gathered} \frac{z}{\overline{z}}=\frac{-5-12i}{13} \\ \frac{z}{\overline{z}}=\frac{x^2+2xyi-y^2}{x^2=y^2} \end{gathered}$$