Ensemble of periodic functions orthogonal to their derivative

Question

What is the most general form of real periodic functions orthogonal to their derivative ?

Trigonometric functions are an obvious example of such functions (e.g. for $f(x)=\sin(x)$, then $\langle f,f'\rangle=\int_{-\infty}^{+\infty}\sin(x) \cos(x) dx=0$), but are these the only periodic functions orthogonal to their derivative ?

Conversely, does the ensemble of periodic functions not orthogonal to their derivative have a name, let alone known generic properties ?

Answer 1

Actually, this is the case for any periodic function. Consider $f(x)$ being periodic with period $T$. That is, $f(x+T)=f(x)$. In particular, $f(T)=f(0)$. Then, integrating by parts,

$$\int\limits_0^Tf(x)f'(x)dx = f(x)f(x)\big|_0^T-\int\limits_0^Tf'(x)f(x)dx$$

Thus,

$$2\int\limits_0^Tf(x)f'(x)dx = f(x)f(x)\big|_0^T = f(T)^2-f(0)^2=0$$

Hence

$$\int\limits_0^Tf(x)f'(x)dx = 0$$