Is the quotient of periodic functions also periodic?

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SUMMARY

The discussion centers on proving that if a function f is periodic with period P, then the function 1/f is also periodic with the same period P. The participants clarify that the proof requires demonstrating that 1/f(s + P) = 1/f(s). Additionally, the discussion introduces a second problem regarding the quotient of two periodic functions, f and g, which may also be periodic, but with a period that could be smaller than P. Examples illustrating this concept are requested.

PREREQUISITES
  • Understanding of periodic functions and their properties
  • Basic knowledge of function transformations
  • Familiarity with mathematical proofs and logical reasoning
  • Concept of function quotients and their implications
NEXT STEPS
  • Study the properties of periodic functions in detail
  • Learn how to prove properties of function transformations
  • Explore examples of periodic functions and their quotients
  • Investigate cases where the period of f/g is less than P
USEFUL FOR

Students studying mathematics, particularly those focusing on calculus and analysis, as well as educators seeking to enhance their understanding of periodic functions and their properties.

Kyyzen
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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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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.
 

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