Iterating the Function ( f(x) e^{-x^2} ) to Infinity: Graph and Convergence

In mathematics, particularly in the realm of functional analysis, it's fascinating to examine the behavior of an infinite iteration of a function. This article will explore how to graph the function ( f^{infty}(x) ) when ( f(x) e^{-x^2} ).

Understanding the Infinite Iteration

To start, let's consider the function ( f(x) e^{-x^2} ). The task is to determine the behavior of the infinite iteration, ( f^{infty}(x) ). One of the key observations is that, after an infinite number of iterations, adding one more iteration doesn't change the value. Mathematically, we can express this as:

Observation 1:
( f^{infty}(x) f(f^{infty}(x)) ) or ( f^{infty}(x) f(x) ).

Therefore, for any ( y f^{infty}(x) ), we must have:

Equation 1: y f(y) e^{-y^2}

Solving the Fixed Point Equation

The equation ( y e^{-y^2} ) looks challenging at first glance, but let's break it down step-by-step.

Step-by-Step Analysis:

Range of ( e^{-x^2} ): Contraction Mapping Principle: The function ( f(x) e^{-x^2} ) is a contraction mapping. This means that it brings points closer together. Formally, for any ( x_1, x_2 in mathbb{R} ), we have: Convergence to a Fixed Point:

Key Insight: Since ( e^{-x^2} ) is a contraction, it has a unique fixed point. This fixed point can be found by iterating ( f(x) e^{-x^2} ) starting from any initial value in ( mathbb{R} ).

Graphing the Convergent Function

To graph ( y f^{infty}(x) ), we need to find the fixed point of the equation ( y e^{-y^2} ). Here's how you can do it on a calculator:

Choose an initial value, such as 0.5. Iterate the function by squaring the value, negating it, and then raising ( e ) to that power. Repeat this process until the values converge. By iterating, you will find that the fixed point is approximately 0.652918...

The graph of ( y f^{infty}(x) ) is simply a horizontal line at ( y 0.652918 ldots ).

Animation and Visualization

To better understand the convergence, you can visualize the process on a graphing calculator or programming environment. Plot the function ( f^{infty}(x) e^{-x^2} ) and observe how the values converge towards the fixed point.

Conclusion

In conclusion, iterating the function ( f(x) e^{-x^2} ) infinitely results in a fixed point that can be found through an iterative process. The function's contraction property ensures that this fixed point is unique and the values converge to it.

Understanding this concept not only provides insight into the behavior of iterative functions but also illustrates the powerful principles of contraction mappings and fixed points in analysis.