Find the first three iterates of the function f(z) = 0.5z + (4 – 2i) for each initial value

Find the first three iterates of the function f(z) = 0.5z + (4 – 2i) for each initial value.

The Correct Answer and Explanation is :

To find the first three iterates of the function ( f(z) = 0.5z + (4 – 2i) ) for a given initial value ( z_0 ), we apply the function iteratively:

1. First iterate: ( z_1 = f(z_0) = 0.5z_0 + (4 – 2i) )
2. Second iterate: ( z_2 = f(z_1) = 0.5z_1 + (4 – 2i) )
3. Third iterate: ( z_3 = f(z_2) = 0.5z_2 + (4 – 2i) )

Let’s compute these iterates for a specific initial value, say ( z_0 = -8 ):

1. First iterate:
   [
   z_1 = 0.5(-8) + (4 – 2i) = -4 + 4 – 2i = -2i
   ]
2. Second iterate:
   [
   z_2 = 0.5(-2i) + (4 – 2i) = -i + 4 – 2i = 4 – 3i
   ]
3. Third iterate:
   [
   z_3 = 0.5(4 – 3i) + (4 – 2i) = 2 – 1.5i + 4 – 2i = 6 – 3.5i
   ]

Thus, the first three iterates for ( z_0 = -8 ) are:

• ( z_1 = -2i )
• ( z_2 = 4 – 3i )
• ( z_3 = 6 – 3.5i )

Explanation:

The function ( f(z) = 0.5z + (4 – 2i) ) is a linear transformation in the complex plane. Each iteration involves scaling the current value ( z ) by 0.5 and then translating it by the complex constant ( 4 – 2i ).

Behavior Analysis:

To understand the behavior of this iterative process, consider the fixed point of the function. A fixed point ( z^* ) satisfies ( f(z^) = z^ ):

[
z^* = 0.5z^* + (4 – 2i)
]

Solving for ( z^* ):

[
z^* – 0.5z^* = 4 – 2i
]
[
0.5z^* = 4 – 2i
]
[
z^* = 8 – 4i
]

This fixed point ( z^* = 8 – 4i ) is where the function maps ( z ) to itself. The iterative process will move any initial value ( z_0 ) closer to this fixed point with each iteration. This is because the scaling factor 0.5 has an absolute value less than 1, causing the iterations to contract towards the fixed point.

General Case:

For any initial value ( z_0 ), the iterates can be computed as follows:

1. First iterate:
   [
   z_1 = 0.5z_0 + (4 – 2i)
   ]
2. Second iterate:
   [
   z_2 = 0.5z_1 + (4 – 2i) = 0.5[0.5z_0 + (4 – 2i)] + (4 – 2i) = 0.25z_0 + 0.5(4 – 2i) + (4 – 2i)
   ]
   Simplifying:
   [
   z_2 = 0.25z_0 + 2 – i + 4 – 2i = 0.25z_0 + 6 – 3i
   ]
3. Third iterate:
   [
   z_3 = 0.5z_2 + (4 – 2i) = 0.5[0.25z_0 + 6 – 3i] + (4 – 2i) = 0.125z_0 + 0.5(6 – 3i) + (4 – 2i)
   ]
   Simplifying:
   [
   z_3 = 0.125z_0 + 3 – 1.5i + 4 – 2i = 0.125z_0 + 7 – 3.5i
   ]

As the iterations proceed, the coefficient of ( z_0 ) decreases geometrically (0.5, 0.25, 0.125, …), and the constant terms approach the fixed point ( 8 – 4i ). Therefore, regardless of the initial value ( z_0 ), the iterates ( z_n ) will converge to the fixed point ( 8 – 4i ) as ( n ) approaches infinity.

This behavior is characteristic of linear functions with a contraction factor (in this case, 0.5), leading to convergence towards a fixed point in the complex plane.