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

Do we have to assume that exists for all ?

Alt sol:

= – +

= IFF =

= =

Hi Jamie, Nice solution. I did it similarly to RoRi but this is a very elegant approach.

Hi jamiehlusko,

I think your solution is wrong because you don’t know if a0 so you are not allowed to use the translation property of integrals.

I checked the book now and you are right.