I am having trouble assimilating periodic function. Let me tell you, I have had a semester of fourier analysis already but reviewing the first chapter got me confused on a trivial equation.
A function is periodic if it satisfies: $x(t) = x(t+T_0)$, where smallest $T_0$ is the fundamental period.
Lets take, for instance, $\sin(x)$. I can intuitively see that period is $2\pi$ BUT $\sin(0)=\sin(\pi)$ as well. By that I mean, the value of $\sin(x)$ is the same at $x=0$ and $x=pi$, then shouldn't $T_0$ be $\pi$ because $T_0$ is fundamental for smallest value of $T_0$ which i just showed is $1$ and NOT $2$.
please help me on this trivial issue.
Best Answer
I think that where you're confused is in the fact that the period is the least number T so that for all x , we have $$ f(x+T)=f(x)$$. For some x (like, in your example of sinx, $sin(0)=sin(\pi)$, it may be the case that there is some $T' < T$ so that$ f(x+T')=f(x)$, but you must have a $T$ that works for all x.