Exploring the Completeness of the Metric Arctanx - Arctany

The Arctan function, a fundamental component in various mathematical applications, possesses unique properties and behaviors across different domains. A critical aspect to investigate pertains to the completeness of the metric Arctan(x) - Arctan(y). This article will delve into how to demonstrate that this metric is not complete when applied to the space from negative infinity to positive infinity, denoted as (-infty, infty). We will also explore how extending the space to include the points at infinity ensures the completeness of this metric.

Understanding the Arctan Metric

The Arctan metric is defined as the difference between the arctangent values of two points x and y. The function arctan(x) maps real numbers to the interval (-frac{pi}{2}, frac{pi}{2}). When two points x_n and x_m are sufficiently far apart, the distance arctan(x_n) - arctan(x_m) approaches zero. In this article, we will demonstrate how this metric fails to meet the completeness requirement within the standard space, but holds when extended to include the points at infinity.

Step 1: Showing Incomplete Metric on (-infty, infty)

To show that the metric arctan(x) - arctan(y) is not complete, we need to establish the existence of a Cauchy sequence that does not converge within the same space.

Defining the Cauchy Sequence

Consider the sequence defined as x_n n for n 1, 2, 3, ldots. This sequence diverges to positive infinity as n increases. To evaluate the metric, we need to consider the distance between two points in the sequence x_n and x_m using the metric arctan(n) - arctan(m).

Evaluating the Metric

As both n and m grow large, the values of arctan(n) and arctan(m) tend toward frac{pi}{2}. Specifically, for sufficiently large n and m, the difference arctan(n) - arctan(m) approaches zero. This indicates that the sequence x_n is indeed a Cauchy sequence in the metric induced by arctan(x) - arctan(y).

Convergence Issue

However, as n to infty, arctan(n) to frac{pi}{2}, which lies outside the original space (-infty, infty). Therefore, the sequence x_n does not converge to any point within (-infty, infty). This shows that arctan(x) - arctan(y) is not a complete metric on (-infty, infty).

Step 2: Extending the Metric to [-infty, infty]

To extend the metric arctan(x) - arctan(y) to a complete metric on [-infty, infty], we redefine the space to include the points at infinity.

Including Points at Infinity

Define a new space [-infty, infty] where we include the points -infty and infty.

Evaluating Behavior at Infinity

In this extended space, the behavior of the Arctan function as x approaches -infty and infty is critical. Specifically,

- arctan(-infty) -frac{pi}{2}

- arctan(infty) frac{pi}{2}

For the same Cauchy sequence x_n n, as n increases, arctan(n) approaches frac{pi}{2}. In the extended space, we can assert that the sequence x_n converges to infty. Similarly, a sequence like x_n -n converges to -frac{pi}{2} and approaches -infty.

Completeness

Since every Cauchy sequence in this extended space converges to a limit in [-infty, infty], we can conclude that the metric arctan(x) - arctan(y) is a complete metric on [-infty, infty].

Conclusion

In summary, the metric arctan(x) - arctan(y) fails to be complete on the space (-infty, infty) because it does not ensure that all Cauchy sequences converge within this space. By extending the space to include the points at infinity, we can ensure that every Cauchy sequence converges, making arctan(x) - arctan(y) a complete metric on [-infty, infty].