Q57 P - Mathematical Methods in Physical Sciences

Question

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

 $\mathit{R}\mathit{e}\mathbf{\left(}{\mathit{z}}^{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{4}$. 

Step-by-Step Solution

Answer

This is an equation of a hyperbola.

Step 1: Given Information

The equation is $\mathrm{Re}\left(z^2\right)=4$.

Step 2: Definition of the Complex number

A complex number can be represented 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  $\sqrt{\mathbf{-}\mathbf{1}}$.

 

The modulus of a complex number can be calculated as:

 

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

Step 3: Find the value

The equation is $\mathrm{Re}\left(z^2\right)=4$.

 

The complex number is $z=x+yi$.

 

$\begin{array}{l}{z}^2={(x+yi)}^2\\ z^2=x^2-y^2+2xyi\\ \mathrm{Re}\left({\mathrm{z}}^2\right)=4\\ x^2-y^2=4\end{array}$ 

 

Hence, this is an equation of a hyperbola.