Let be an odd, periodic, integrable everywhere function with period 2. Then, define
- Prove that for all .
- Prove that is an even, periodic function with period 2.
- Proof. We can establish this by a direct computation.
But then we have
Since we then have the requested result,
- First, to show that is even we need to show for all . We compute, using the fact that is odd and the expansion/contraction property of the integral (Theorem 1.19 in Apostol) with .
Next, to show that is periodic with period 2 we must show for all .