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,