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
.