How to find fundamental period of the sum of two periodic functions

Question

Suppose we have a function $h(x) = f(x) + g(x)$ where $f$ and $g$ are periodic functions with fundamental period $T_1$ and $T_2$ respectively. We know that the least common integral multiple of $T_1$ and $T_2$, say $T$, is the period of $h$ but we don't have surety that $T$ is the fundamental period. For example, take $h(x) = |\sin x| + |\cos x|$. How can we get the fundamental period of $h$?

Proposed Solution: Since we know that if $p$ is the fundamental period of some function then every integral multiple of $p$ is the period of that function. We can reverse this thinking. Since we know that $T$ is the period of h then the possibilities for the fundamental periods are $T, T/2, T/3, T/4,\dots $. The minimum among these possibilities will be the fundamental period of $h$.

Problem in the proposed solution: For example, if we find that $T$ and $T/2$ is the period of $h$ but $T/3$ is not the period of $h$. This could be done by verifying $$ h(x+p)= h(x)\quad \text{for all}\ x \in \text{Dom}(h). $$ How can we say that no other smaller number like $T/4, T/5$ cannot be the period of $h$?

Answer 1

According to the OP and its comments, it seems that the main concern of this post is the following.

Let $f$ and $g$ be periodic functions with fundamental period $T_1$ and $T_2$, respectively. Suppose that the function $h=f+g$ is periodic.^$\dagger$ So, the fundamental period of $h$ must be a submultiple of the least common integral multiple of $T_1$ and $T_2$, say $ \DeclareMathOperator{\lcm}{lcm \, } T= \lcm (T_1, T_2)$. Does there exist a general way to find the minimum natural number $n$ for which one of the submultiples $T$, $\frac{T}{2}$, ... , $\frac{T}{n}$ is the fundamental period of $h$?

The answer to the question is negative. The following examples can give insight about this fact.

• Let $f(x)=3\sin x$ and $g(x)=-4\sin^3 x$; both of them have fundamental period $2\pi$. Now, the function$$h(x)=f(x)+g(x)=3\sin x -4\sin ^3 x=\sin 3x$$has fundamental period $\frac{2\pi }{3}$ (Please note that if we change the coefficients in $f$ or $g$ then we can have a periodic function with fundamental period $2\pi$; for example, $h(x)=2\sin x -4\sin ^3 x$ is periodic with fundamental period $2\pi$).

• Let $f(x)=5\sin x$ and $g(x)=-20\sin^3 x + 16\sin ^5 x$; both of them have fundamental period $2\pi$. Now, the function$$h(x)=f(x)+g(x)=5\sin x -20\sin ^3 x + 16 \sin ^5 x=\sin 5x$$has fundamental period $\frac{2\pi }{5}$ (Please note that if we change the coefficients in $f$ or $g$ then we can have a periodic function with fundamental period $2\pi$; for example, $h(x)=5\sin x -21\sin ^3 x + 16 \sin ^5 x$ is periodic with fundamental period $2\pi$).

$$\vdots$$

• (This can be confirmed graphically for many arbitrarily large values of $n$) Let $f(x)= n \sin x$, $n$ is odd, and $$g(x)=\sum_{k=1}^{\frac{n-1}{2}}(-1)^k n\binom{\frac{n-1}{2}}{k}\sin ^{k+1}x + \sum_{k=1}^{\frac{n-1}{2}}(-1)^k \binom{n}{2k+1}\sin ^{2k+1}x(1-\sin ^2x)^{\frac{n-1}{2}-k};$$both of them have fundamental period $2\pi$. Now, the function$$h(x)=f(x)+g(x)= n \sin x + \sum_{k=1}^{\frac{n-1}{2}}(-1)^k n\binom{\frac{n-1}{2}}{k}\sin ^{k+1}x+ \sum_{k=1}^{\frac{n-1}{2}}(-1)^k \binom{n}{2k+1}\sin ^{2k+1}x(1-\sin ^2x)^{\frac{n-1}{2}-k}= \sin nx$$has fundamental period $\frac{2\pi }{n}$ (Please note that if we change the coefficients in $f$ or $g$ then we can have a periodic function with fundamental period $2\pi$).

Conclusion

By examining the behavior of the above examples, we can conclude that there is no general way to to find the minimum natural number $n$ for which one of the submultiples $T$, $\frac{T}{2}$, ... , $\frac{T}{n}$ is the fundamental period of $h$.

The main point here is that the sum of two functions may behave very differently from each of them. When two periodic functions with some fluctuations in their graphs are added, the fluctuations may be canceled out in the sum of the functions, so the resultant shape of the function may be more symmetric. So, in such cases, the period of the sum of the functions may be reduced to some extent depending on the characteristics of the functions. As it was said in the mentioned examples, making some small change to the coefficients in the given functions may lead to a different fundamental period.

---

Footnote

$\dagger$ Please note that the sum of two periodic functions is not necessarily a periodic function. For more information, please see this post or this post.