Define a function
where denotes the greatest integer less than or equal to , also called the floor function.
Then, define a function
- Draw the graph of for and prove that is periodic with period 1.
- For , prove
and prove that is also periodic with period 1.
- Give a formula for in terms of the floor of .
- Find a such that
- For the constant from part (d), define
Prove that is periodic with period 1 and that if , we have,
- The graph of is as follows:
Then, we prove that is periodic with period 1.
Proof. We compute to show (where I’ve replaced Apostol’s notation with since this less likely to cause confusion, also we use the solution to this exercise in the second line)
- Proof. First, we establish the requested formula,
This was the requested formula. Next, to show is periodic with period 1, we show for any ,
Then, using the formula in the first part of (b) we have
Further, from part (a) we know is periodic with period 1, so , and we have,
by definition of . Thus, is periodic with period
- To express in terms of of we compute as follows:
where since is an integer by definition, and has period 1, so for any integer . Continuing,
Here, we know the term in the integral is zero since for all since . Then,
This is the requested expression of in terms of .
- Here we compute the integral, using the formula for we established in part (c), and solve for the requested constant ,
- First we give the proof that is periodic with period 1.
Proof. We compute,
Next, we establish the requested formula. If , we have