Coprime Commensurate Periods

Oct 13, 2025 by windailys.com 49 views

Introduction

In the realm of periodic functions, understanding the properties of their sums is crucial for various applications in mathematics and physics. When dealing with two functions, $f$ and $g$ , each having a shortest period of $t_1=2$ and $t_2=3$ , respectively, we are naturally inclined to assume that the sum of these functions, $h=f+g$ , would have a shortest period of $6$ . However, this assumption may not always hold true. In this article, we will delve into the intricacies of the shortest period of the sum of two functions with coprime commensurate periods and explore the conditions under which this assumption is valid.

Coprime Commensurate Periods

Before we proceed, let's clarify the concept of coprime commensurate periods. Two numbers, $a$ and $b$ , are said to be coprime if their greatest common divisor (GCD) is $1$ . In the context of periodic functions, the periods $t_1$ and $t_2$ are commensurate if there exists a positive integer $k$ such that $t_1 = k \cdot t_2$ or $t_2 = k \cdot t_1$ . However, when $t_1$ and $t_2$ are coprime, they have no common factors other than $1$ , and their ratio is an irrational number.

The Shortest Period of the Sum

Given two functions, $f$ and $g$ , with shortest periods $t_1=2$ and $t_2=3$ , respectively, we want to determine the shortest period of their sum, $h=f+g$ . To do this, we need to consider the properties of periodic functions and their sums.

A function $f$ is said to be periodic with period $T$ if $f(x+T) = f(x)$ for all $x$ in the domain of $f$ . The shortest period of a function is the smallest positive value of $T$ for which this property holds.

When we add two periodic functions, $f$ and $g$ , with periods $t_1$ and $t_2$ , respectively, the resulting function, $h=f+g$ , may not necessarily have a period that is a multiple of both $t_1$ and $t_2$ . In fact, the period of $h$ can be a divisor of both $t_1$ and $t_2$ .

Counterexample

To illustrate this, let's consider a simple counterexample. Suppose we have two functions, $f(x) = \sin(\frac{\pi x}{2})$ and $g(x) = \sin(\frac{\pi x}{3})$ . The shortest periods of $f$ and $g$ are $t_1=2$ and $t_2=3$ , respectively. Now, let's consider the sum of these functions, $h=f+g$ .

Using the angle addition formula for sine, we can rewrite $h$ as:

$h(x) = \sin(\frac{\pi x}{2}) + \sin(\frac{\pi x}{3}) = 2 \sin(\frac{\pi x}{6}) \cos(\frac{\pi x}{6})$

The period of $h$ is $6$ , which is the least common multiple (LCM) of $t_1=2$ and $t_2=3$ . However, this is not the only possible period of $h$ . In fact, $h$ also has a period of $12$ , which is a multiple of both $t_1$ and $t_2$ .

Conclusion

In conclusion, the assumption that the sum of two functions with coprime commensurate periods has a shortest period equal to the least common multiple of their periods is not always true. The period of the sum can be a divisor of both periods, and it may not necessarily be the least common multiple. This highlights the importance of carefully analyzing the properties of periodic functions and their sums.

Open Questions

While we have explored the properties of the sum of two functions with coprime commensurate periods, there are still many open questions in this area. For example:

What are the conditions under which the sum of two functions with coprime commensurate periods has a shortest period equal to the least common multiple of their periods?
Can we generalize the results to the sum of more than two functions?
How do the properties of the sum of periodic functions relate to other areas of mathematics, such as number theory and algebra?

These questions highlight the need for further research in this area and provide a rich source of problems for mathematicians and physicists to explore.

References

[1] A. Erdélyi, "Tables of Integral Transforms", McGraw-Hill, 1954.
[2] E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis", Cambridge University Press, 1927.
[3] H. M. Edwards, "Riemann's Zeta Function", Academic Press, 1974.

Q: What is the shortest period of the sum of two functions with coprime commensurate periods?

A: The shortest period of the sum of two functions with coprime commensurate periods is not necessarily the least common multiple (LCM) of their periods. In fact, the period of the sum can be a divisor of both periods.

Q: Can you provide an example to illustrate this?

A: Consider two functions, $f(x) = \sin(\frac{\pi x}{2})$ and $g(x) = \sin(\frac{\pi x}{3})$ . The shortest periods of $f$ and $g$ are $t_1=2$ and $t_2=3$ , respectively. The sum of these functions, $h=f+g$ , has a period of $6$ , which is the LCM of $t_1$ and $t_2$ . However, $h$ also has a period of $12$ , which is a multiple of both $t_1$ and $t_2$ .

Q: What are the conditions under which the sum of two functions with coprime commensurate periods has a shortest period equal to the least common multiple of their periods?

A: Unfortunately, there are no known conditions that guarantee the sum of two functions with coprime commensurate periods will have a shortest period equal to the least common multiple of their periods. The example provided earlier shows that this is not always the case.

Q: Can we generalize the results to the sum of more than two functions?

A: While we have explored the properties of the sum of two functions with coprime commensurate periods, there is no known generalization to the sum of more than two functions. The properties of the sum of periodic functions are complex and depend on the specific functions involved.

Q: How do the properties of the sum of periodic functions relate to other areas of mathematics, such as number theory and algebra?

A: The properties of the sum of periodic functions are closely related to other areas of mathematics, such as number theory and algebra. For example, the study of periodic functions is closely tied to the study of modular forms, which are functions on the upper half-plane of the complex numbers that are periodic with respect to the lattice of complex numbers with integer real and imaginary parts.

Q: What are some open questions in this area?

A: Some open questions in this area include:

What are the conditions under which the sum of two functions with coprime commensurate periods has a shortest period equal to the least common multiple of their periods?
Can we generalize the results to the sum of more than two functions?
How do the properties of the sum of periodic functions relate to other areas of mathematics, such as number theory and algebra?

Q: What are some potential applications of this research?

A: The study of the properties of the sum of periodic functions has potential applications in various areas, including:

Signal processing: The study of periodic functions is closely related to the study of signal processing, which is a crucial area of research in electrical engineering and computer science.
Number theory: The study of periodic functions is closely related to the study of number theory, which is a fundamental area of mathematics that has numerous applications in cryptography and coding theory.
Algebra: The study of periodic functions is closely related to the study of algebra, which is a fundamental area of mathematics that has numerous applications in computer science and engineering.

Q: What are some potential future directions for this research?

A: Some potential future directions for this research include:

Developing new techniques for analyzing the properties of the sum of periodic functions.
Exploring the connections between the properties of the sum of periodic functions and other areas of mathematics, such as number theory and algebra.
Developing new applications for the study of periodic functions in various areas, including signal processing, number theory, and algebra.

References

[1] A. Erdélyi, "Tables of Integral Transforms", McGraw-Hill, 1954.
[2] E. T. Whittaker and G. N. Watson, "A Course of Modern Analysis", Cambridge University Press, 1927.
[3] H. M. Edwards, "Riemann's Zeta Function", Academic Press, 1974.

Note: The references provided are a selection of classic texts in the field of mathematics and physics. They are not necessarily directly related to the topic of the shortest period of the sum of two functions with coprime commensurate periods.