Q54P - Mathematical Methods in Physical Sciences

Question

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

 $\left|z-1\right|\mathbf{<}\mathbf{1}$. 

Step-by-Step Solution

Answer

This is the equation of the circle with center $\left(1,0\right)$ and radius $r=1$  except for the outer border.

Step 1: Given Information

The equation is $\left|Z-1\right|<1$.

Step 2: Definition of the Complex number

A complex number can be written 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 $\left|z-1\right|<1$.

 

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

 

$\left|x+yi-1\right|<1$ 

 

Solve further as:

 

$\begin{array}{l}\sqrt{{(x-1)}^{{}^2}+y^2}<1\\ (x-1{\}}^2+y^2<1\end{array}$ 

 

Hence, this is the equation of the circle with center $\left(1,0\right)$ and radius $r=1$ except for the outer border.