Q64P - Mathematical Methods in Physical Sciences

Question

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

 ${\mathit{z}}^{\mathbf{2}}\mathbf{=}\mathbf{-}{\overline{\mathbf{z}}}^{\mathbf{2}}\mathbf{.}$. 

Step-by-Step Solution

Answer

The equation is $y=\pm x$  , the straight lines. 

Step 1: Given Information

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

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 $\sqrt{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}\mathbf{ }\mathbf{.}}$  .

The modulus of a complex number can be calculated as:

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

Step 3: Find the value.

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

The complex number is given below as:

 $$\begin{gathered} z^2={(x+yi)}^2 =x^2-y^2+2xyi \\ {z}^2={(x+yi)}^2 \\ =x^2-y^2+2xyi \end{gathered}$$

 

Evaluate the value:

$$\begin{gathered} x^2-y^2+2xyi=-(x^2-y^2-2xyi) \\ x^2=y^2 \\ y=\pm x \end{gathered}$$ 

Hence, the equation is $y=\pm x$ , the straight lines.