Home » Periodic Functions

# Prove some properties of even, periodic, integrable functions

Let be an integrable, even, periodic function with period 2. Define

1. Prove is odd and .
2. Compute and using that .
3. Find the value of that makes periodic with period 2.

1. Proof. Using the expansion/contraction property (Apostol, Thm 1.19 with ) we compute,

Thus, is odd. Next,

2. 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

3. 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,

# Prove some formulas for odd periodic functions

Let be an odd, periodic, integrable everywhere function with period 2. Then, define

1. Prove that for all .
2. Prove that is an even, periodic function with period 2.

1. Proof. We can establish this by a direct computation.

But then we have

Since we then have the requested result,

2. 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 .

# Prove an integral formula for periodic functions

Let be an integralable function on , and let be periodic with period . Prove

for all .

Proof. For any with we know from a previous exercise that there exists a unique such that

Then, we start by splitting the integral into two pieces. (The goal here is to rearrange the integral so that it starts at and ends at , and then use the fact that is periodic to conclude that this integral is the same as the one from 0 to .)

This completes the proof