Is the quotient of periodic functions also periodic?

In summary, the conversation discusses how to prove that 1/f is periodic with period P, given that f is also periodic with period P. The attempt at a solution involves using the equation for a periodic function, but it is determined that more steps are needed to prove the 1/f part. The second question also relates to periodic functions, but it asks to prove that the quotient of two periodic functions can have a period smaller than P. An example is also requested to illustrate this concept.
  • #1
Kyyzen
1
0

Homework Statement


Let f be periodic with period P. Prove that 1/f is periodic with period P.


The Attempt at a Solution



f(s+P)= f(s) I know that is the equation for a periodic function. I am not sure how to
prove the 1/f part though. Would I just do this:

1/f(s+P)=f(s)? I'm just not sure what exactly to do. Please help =D haha.

*ALSO* I have another question that states.
Let f, g be periodic with period P. Prove that f/g is periodic, but the period could be smaller than P. In addition, give an example that illustrates this.

I have not attempted this one yet.
 
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  • #2
Kyyzen said:

Homework Statement


Let f be periodic with period P. Prove that 1/f is periodic with period P.


The Attempt at a Solution



f(s+P)= f(s) I know that is the equation for a periodic function. I am not sure how to
prove the 1/f part though. Would I just do this:

1/f(s+P)=f(s)?
No. Don't you need to show that 1/f(s + P) = 1/f(s)?
Kyyzen said:
I'm just not sure what exactly to do. Please help =D haha.

*ALSO* I have another question that states.
Let f, g be periodic with period P. Prove that f/g is periodic, but the period could be smaller than P. In addition, give an example that illustrates this.

I have not attempted this one yet.
 

Related to Is the quotient of periodic functions also periodic?

1. What is a periodic function?

A periodic function is a mathematical function that repeats its values at regular intervals. This means that as the independent variable (often represented by x) increases, the function will eventually return to the same output values it had before. This repetition creates a pattern that is characteristic of periodic functions.

2. What are some examples of periodic functions?

Some common examples of periodic functions include sine, cosine, and tangent functions. These are also known as trigonometric functions. Other examples include square waves, sawtooth waves, and triangular waves.

3. How can I recognize a periodic function from its graph?

A periodic function will have a repeating pattern in its graph. This pattern may be symmetrical or asymmetrical, but it will always repeat itself at regular intervals. Additionally, the graph will extend infinitely in both the positive and negative directions.

4. How do I find the period of a periodic function?

The period of a periodic function is the length of one complete cycle of the function's graph. To find the period, you can look for the distance between two consecutive peaks or troughs of the graph. Alternatively, you can use the formula T = 2π/b, where b is the coefficient of x in the function's equation.

5. How are periodic functions used in real life?

Periodic functions have many real-world applications, including in physics, engineering, and economics. They can be used to model and predict phenomena that occur in cycles, such as the motion of a pendulum, the fluctuation of stock prices, or the behavior of waves. They are also used in signal processing and data analysis to identify repeating patterns and trends.

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