I'm wondering, is the function $f=(\sin{x})(\sin{\pi x})$ is periodic?
My first inclination would be two assume that if the periods of the individual sine expressions, $p_1 \text{and}\space p_2$ have the quality that $p_1 \times a = p_2 \times b$ where $a \space\text{and}\space b$ are integers, then the entire function will eventually repeat after a period of $p_1 \times a$.
If that is true, than I think $f$ might not be periodic due to the fact that two Pi is irrational.
Does anyone know the answer and/or weather my thinking is correct? I've never seen a function like this before, so I'm really curious.
Best Answer
Using the product-to-sum formula, $$ \sin x \sin\pi x = \frac{1}{2} \left( \cos ((1-\pi)x) - \cos((1 + \pi)x) \right) $$ but $\frac{1-\pi}{1+\pi} \not\in \mathbb{Q}$ and $\sin x \sin \pi$ is continuous, so this function is not periodic.