Properties of periodic functions

Question

Let $f$ and $g$ be periodic functions of period $p$. Then $af(x)+bf(x)$ with $a,b$ constants and $f(x)g(x)$ are both of period $p$

I'm not exactly sure how to prove these properties of periodic functions. I think I may have proven the first one, but as I remarked, I'm not sure.

To prove the first one, $f(x)=f(x+p)$. So $a f(x+p)+bg(x+p)=af(x)+bg(x)$.

I think you really meant $af(x)+bg(x)$, right? – MPW Mar 02 '14 at 21:16 — MPW, Mar 02 '14 at 21:16
No, $a$ is just in front of $f(x)$. – emka Mar 02 '14 at 22:21 — emka, Mar 02 '14 at 22:21

score 1 · Accepted Answer · answered Mar 02 '14 at 21:14

1

The same approach works for the other example: $f(x+p)g(x+p)=f(x)g(x+p)=f(x)g(x)$. The first equality follows from the $p$-periodicity of $f$, the second from that of $g$.

answered Mar 02 '14 at 21:14

MPW

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Properties of periodic functions

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