Finding the period of sum of two periodic functions.

Many a times it is asked to find the period of combination of two functions given the period of individual functions.

Let’s take this example:

If $$f(x)= \cos ax + \sin x$$ is periodic, then $$a$$ cannot be? $$\pi$$, 0.3, 0.5, 5

I can see that the period of $$\cos ax$$ is $$\frac{2\pi}{a}$$ and that of $$\sin x$$ is $$\pi$$ but how to find the period of $$f(x)$$?

Another example of very similar concept is

If $$f(x)$$ and $$g(x)$$ are periodic functions with period $$7$$ and $$11$$ respectively. Then the period of $$F(x) = f(x) g(x/5) - g(x) f(x/3)$$

I know the period of $$g(x/5)$$ is $$11/5$$ and that of $$f(x/3)$$ is $$7/3$$, but how to combine the periods?

• The keyword is commensurate. Commented Sep 7, 2020 at 8:26
• @GerryMyerson Can you please explain me a situation where the individual periods are incommensurate? Commented Sep 7, 2020 at 10:09
• The period of $\cos\pi x$ is incommensurate with the period of $\sin x$. Commented Sep 7, 2020 at 12:26
• If the period of $f(x)$ is $7$, then the period of $f(x/3)$ is not $7/3$, it's $7\times3=21$. Commented Sep 8, 2020 at 11:20

The period of a function is the smallest nonzero $$t$$ such that

$$f(x+t)=f(x)$$ and by induction $$f(x+kt)=f(x).$$

Now if $$f$$ is a combination of two functions of known and commensurate periods $$t',t''$$, there are two relatively prime integers such that

$$k't'=k''t''=t.$$

and we have

$$g(x+k't')=g(x),\\h(x+k''t'')=h(x)$$ which shows that $$f$$ is certainly periodic with a period not exceeding $$t$$.

On the other hand, if $$t',t''$$ are incommensurate, $$f$$ is aperiodic.

Anyway, the period can be shorter, and this depends on the particular combination of $$g,h$$ defining $$f$$. Unfortunately, I know of no way to find the shorter period when there is one, in another way than... looking for the period. E.g. $$\sin 5x+\cos x$$ and $$\sin 5x-\cos x$$ are both of period $$2\pi$$, but their sum $$2\sin 5x$$ has the period $$\dfrac{2\pi}5$$.

• It's really a very nice answer, very well explained. So, for $\cos ax + \sin x$, the individual periods are $\frac{2\pi}{a}$ and $2 \pi$, how to find $k'$ and $k''$? I think we do something like this $$a ~ \frac{2\pi}{a} = 1\times 2\pi = 2\pi$$ but $a$ and $1$ are not co-prime. Commented Sep 7, 2020 at 10:07
• @Knight: $k',k''$ can be any relatively prime integers and $a=\dfrac{k'}{k''}$. The excluded value is obvious.
– user65203
Commented Sep 7, 2020 at 10:11
• Can you please stress a little more on why the function will be aperiodic when the ration of periods of individual functions is not a rational number? Commented Sep 7, 2020 at 10:14
• @Knight: solve $k't'=k''t''$.
– user65203
Commented Sep 7, 2020 at 10:17
• $$\frac{k’}{k’’}= \frac{t’}{t’’}$$ Commented Sep 7, 2020 at 10:24