# Integral proof involving periodic functions

Guys my head is gonna explode with this problem... i have been trying to solve it for the past 2 weeks non-stop... basically it's a problem involving integration and periodic functions. the problem states the following:

If f is a periodic function with its period a, prove that: $$\int _0^{+\infty}\:\frac{f\left(x\right)}{x}dx=\frac{\pi }{a}\int _0^{\frac{a}{2}}\:\frac{f\left(x\right)}{tan\left(\frac{\pi x}{a}\right)}dx$$

I've been given this problem from a friend(he doesn't know how to solve it either..) and, I hope that you guys help me with this it because i do not even know how to begin...

• Have you tried any substitutions for $x$? – астон вілла олоф мэллбэрг Aug 18 '16 at 0:05
• what is the original source of the problem? Note that it is nonsense as written, neither side converges for arbitrary $f$ – Will Jagy Aug 18 '16 at 0:11
• This can't be true for arbitrary function. Suppose $f(x)=g(x)$ for $0\leq x\leq a/2$ and $f(x)=h(x)$ for $a/2\leq x\leq a$ for some function $g, h$. Then LHS depends on $h$ but RHS doesn't. – Seewoo Lee Aug 18 '16 at 0:22