Q63P - Mathematical Methods in Physical Sciences

Question

Question: Describe geometrically the set of points in the complex plane satisfying the following equations.  .

Step-by-Step Solution

Answer

The equation is for coordinate axes.

Step 1: Given Information

The equation is $z^2={\overline{z}}^2$ .

Step 2: Definition of the Complex number

A complex number can be shown as:

$\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}\mathit{i}\mathit{y}$  

Where z is the complex number, x and y are real numbers, and i is known as iota, whose value is  $\sqrt{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{ }}$ .

 

The modulus of a complex number can be calculated as:

 $\mathbf{\left|}\mathit{z}\mathbf{\right|}\mathbf{=}\mathbf{\surd }\mathbf{(}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{)}$ 

Step 3: Find the value

The equation is $z^2={\overline{z}}^2$ .

 

The complex number is $z^2={\overline{z}}^2$  .

 $$\begin{gathered} {\left(x+iy\right)}^2={\left(x+iy\right)}^2 \\ {x}^{2 }+2ixy-y^2=x^{2 }+2ixy-y^2 \\ 4ixy=0 \end{gathered}$$

If x = 0, then y will be a real number and if y = 0, then x will be a real number.

 

Hence, the equation is for coordinate axes.