# The sum and multiplication of periodic functions

## Homework Statement

Hi, my question is whether the sum and multiplication of two periodic functions (with a common period) are periodic.
Our functions are R$\rightarrow$R.

## The Attempt at a Solution

f(x)=f(x+T) g(x)=g(x+T) T is the period.
h(x)=f(x)+g(x)
h(x+T)=f(x+T)+g(x+T)=f(x)+g(x)=h(x)

Hence, h(x) is also periodic.
What I did is similar for multiplication. Is there any flaw in this? I searched a bit and found out this may not hold every time, but I guess that was about Fourier series, which I have no idea about.
Thanks for any hint :)

## The Attempt at a Solution

LCKurtz
Homework Helper
Gold Member
Your argument is fine. You only potentially run into difficulties when the periods aren't the same.

Thanks, LCKurtz. Could you please explain what happens if the periods are not the same? Is it too complicated for a freshman in maths? :)

LCKurtz
Homework Helper
Gold Member
Thanks, LCKurtz. Could you please explain what happens if the periods are not the same? Is it too complicated for a freshman in maths? :)

If you have periods like 2 and 3, then you need to use the least common multiple for the period of the sum. But if your two periods are 2 and $\pi$, there is no lcm and the sum isn't periodic at all. That happens when the ratio of the periods isn't a rational number.

Thanks again :) Wow, I haven't thought of it before...
One last question, if you don't mind me :) What is an almost periodic function? It is a term I came across today, and would be grateful if you could explain this, too.

LCKurtz