Doubt in finding periods of sum of two functions

Question

It has been said that the period of sum of two functions f and g is the LCM of their periods if they exist. Further it has been given that even if LCM exist, it need not be the fundamental period.

Then, is there any algorithm to find the fundamental period, other than hit and trial method?

Please guide me in this regard.

Answer 1

Let, $h(x)=f(x)+g(x)$

Let the fundamental periods of $f$ and $g$ be $T_1$ and $T_2$ respectively.

To find the fundamental period of $h(x)=f(x)+g(x)$, find the L.C.M. of $T_1$ and $T_2$.

Let L.C.M. ($T_1$,$T_2$) $= T$

If L.C.M exists then $h$ is periodic with $T$ as a period but $T$ need not be fundamental. So check whether the submultiples of $T$ are satisfying the condition for periodicity.

For example, if $h(x+\frac{T}{2})=h(x)$ then check whether $h(x+\frac{T}{4})=h(x)$. If yes then proceed finding further submultiples of $T$. If not then $\frac{T}{2}$ is the fundamental period.