Let be an integrable, even, periodic function with period 2. Define
- Prove
is odd and
.
- Compute
and
using that
.
- Find the value of
that makes
periodic with period 2.
- Proof. Using the expansion/contraction property (Apostol, Thm 1.19 with
) we compute,
Thus,
is odd. Next,
- First, we compute
,
where the second to last line followed from the fact that
is even and this exercise, so
.
Since
by definition, the above also gives us
. So, again using part (a) (this time with
) we have
- Here we want to find a value of
such that
is periodic with period 2. For
to be periodic with period 2, we must have
But, since
, and the above must be true for all
, we evaluate at
and use part (b) that gave us
to compute,