Counterexamples and Generalizations

Oct 13, 2025 by windailys.com 59 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, a natural question arises: does the sum of these functions, $h=f+g$ , necessarily have a shortest period of $6$ ? In this article, we delve into the intricacies of this problem, exploring the relationship between the periods of individual functions and their sum.

Background and Notation

Before we proceed, let's establish some notation and background information. 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. Two numbers are coprime if their greatest common divisor is $1$ . In this context, we are dealing with two functions, $f$ and $g$ , with coprime periods $t_1=2$ and $t_2=3$ , respectively.

It is Not Generally True

As hinted in the problem statement, it is not generally true that the sum of two functions with coprime periods will have a shortest period equal to the product of their individual periods. To illustrate this, consider the following example:

Example 1

Let $f(x) = \sin(\frac{\pi x}{2})$ and $g(x) = \sin(\frac{\pi x}{3})$ . Both $f$ and $g$ have periods of $2\pi$ , but their periods are coprime. The sum of these functions, $h(x) = f(x) + g(x)$ , has a period of $\pi$ , which is not equal to the product of their individual periods, $2\pi \cdot 3\pi = 6\pi$ .

This example demonstrates that the sum of two functions with coprime periods does not necessarily have a shortest period equal to the product of their individual periods.

Counterexamples and Generalizations

To further explore this problem, let's consider some counterexamples and generalizations.

Counterexample 1

Consider the functions $f(x) = \sin(\frac{\pi x}{2})$ and $g(x) = \sin(\frac{\pi x}{3})$ . As mentioned earlier, the sum of these functions, $h(x) = f(x) + g(x)$ , has a period of $\pi$ , which is not equal to the product of their individual periods, $2\pi \cdot 3\pi = 6\pi$ .

Counterexample 2

Consider the functions $f(x) = \cos(\frac{\pi x}{2})$ and $g(x) = \cos(\frac{\pi x}{3})$ . The sum of these functions, $h(x) = f(x) + g(x)$ , has a period of $\frac{2\pi}{\gcd(2,3)} = \frac{2\pi}{1} = 2\pi$ , which is equal to the product of their individual periods, $2\pi \cdot 3\pi = 6\pi$ .

These counterexamples illustrate that the sum of two functions with coprime periods does not necessarily have a shortest period equal to the product of their individual periods.

Generalizations and Open Problems

While the examples and counterexamples provided above demonstrate the complexity of this problem, there are some generalizations and open problems that can be explored.

Generalization 1

If $f$ and $g$ are two functions with coprime periods $t_1$ and $t_2$ , respectively, then the sum of these functions, $h=f+g$ , has a period of $\frac{t_1t_2}{\gcd(t_1,t_2)}$ .

Open Problem 1

Is it true that if $f$ and $g$ are two functions with coprime periods $t_1$ and $t_2$ , respectively, then the sum of these functions, $h=f+g$ , has a shortest period equal to the product of their individual periods, $t_1t_2$ ?

Conclusion

In conclusion, the problem of determining the shortest period of the sum of two functions with coprime commensurate periods is a complex and intriguing one. While there are some generalizations and counterexamples that can be explored, there are still many open problems and questions that remain to be answered. Further research and investigation are needed to fully understand the properties of the sum of periodic functions.

References

[1] Krantz, S. G. (1999). Handbook of Complex Variables. Birkhäuser.
[2] Zorich, V. A. (2004). Mathematical Analysis I. Springer.
[3] Bartle, R. G. (2011). The Elements of Real Analysis. Wiley.

Additional Resources

[1] Wikipedia: Periodic Function. Retrieved from https://en.wikipedia.org/wiki/Periodic_function
[2] MathWorld: Periodic Function. Retrieved from https://mathworld.wolfram.com/PeriodicFunction.html
[3] Khan Academy: Periodic Functions. Retrieved from https://www.khanacademy.org/math/calculus/periodic-functions
Q&A: The Shortest Period of the Sum of $2$ Functions with Coprime Commensurate Periods =====================================================================================

Frequently Asked Questions

In this article, we will address some of the most common questions related to the shortest period of the sum of two functions with coprime commensurate periods.

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

A: The shortest period of the sum of two functions with coprime periods is not necessarily equal to the product of their individual periods. In fact, it can be equal to the product of their individual periods, or it can be a different value altogether.

Q: Can you provide an example of two functions with coprime periods whose sum has a shortest period equal to the product of their individual periods?

A: Yes, consider the functions $f(x) = \cos(\frac{\pi x}{2})$ and $g(x) = \cos(\frac{\pi x}{3})$ . The sum of these functions, $h(x) = f(x) + g(x)$ , has a period of $\frac{2\pi}{\gcd(2,3)} = \frac{2\pi}{1} = 2\pi$ , which is equal to the product of their individual periods, $2\pi \cdot 3\pi = 6\pi$ .

Q: Can you provide an example of two functions with coprime periods whose sum has a shortest period different from the product of their individual periods?

A: Yes, consider the functions $f(x) = \sin(\frac{\pi x}{2})$ and $g(x) = \sin(\frac{\pi x}{3})$ . The sum of these functions, $h(x) = f(x) + g(x)$ , has a period of $\pi$ , which is not equal to the product of their individual periods, $2\pi \cdot 3\pi = 6\pi$ .

Q: Is it true that the sum of two functions with coprime periods always has a shortest period equal to the product of their individual periods?

A: No, it is not true that the sum of two functions with coprime periods always has a shortest period equal to the product of their individual periods. In fact, as demonstrated by the examples above, the shortest period of the sum can be equal to the product of their individual periods, or it can be a different value altogether.

Q: Can you provide a general formula for the shortest period of the sum of two functions with coprime periods?

A: Yes, if $f$ and $g$ are two functions with coprime periods $t_1$ and $t_2$ , respectively, then the sum of these functions, $h=f+g$ , has a period of $\frac{t_1t_2}{\gcd(t_1,t_2)}$ .

Q: Is there an open problem related to the shortest period of the sum of two functions with coprime periods?

A: Yes, there is an open problem related to the shortest period of the sum of two functions with coprime periods. Specifically, it is not known whether the sum of two functions with coprime periods always has a shortest period equal to the product of their individual periods.

Conclusion

References

[1] Krantz, S. G. (1999). Handbook of Complex Variables. Birkhäuser.
[2] Zorich, V. A. (2004). Mathematical Analysis I. Springer.
[3] Bartle, R. G. (2011). The Elements of Real Analysis. Wiley.

Additional Resources

[1] Wikipedia: Periodic Function. Retrieved from https://en.wikipedia.org/wiki/Periodic_function
[2] MathWorld: Periodic Function. Retrieved from https://mathworld.wolfram.com/PeriodicFunction.html
[3] Khan Academy: Periodic Functions. Retrieved from https://www.khanacademy.org/math/calculus/periodic-functions