Q61P - Mathematical Methods in Physical Sciences

Question

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

Step-by-Step Solution

Answer

The equation is the lower part of the xy-plane. 

Step 1: Given Information

The equation is $Imz<0.$  

Step 2: Definition of the Complex number

A complex number can be expressed 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 $\sqrt{\mathbf{(}\mathbf{-}\mathbf{1}\mathbf{)}}$  .

 

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 $Imz<0.$ .

 

The complex number is $z=x+iy$ .

 $$\begin{gathered} Imz<0 \\ Y<0 \end{gathered}$$

Hence, the equation is the lower part of the xy-plane.