- Prove the integral formula,
for integers and .
- Prove the following orthogonality relations of sine and cosine using the relation in part (a), where and are integers with .
- Proof. First, if then we have
If then we have
- Proof. These are all direct computations using part (a). Here they are,
(The final line follows by part (a) and since by hypothesis which implies , and .) Next,
The third formula,
For the next one we use the identities for and derived in this exercise (Section 9.10, Exercise #4(b)).
Finally,
I think there is a typo (or a mistake) in part a), because \int{exp(ikx)dx} = exp(ikx)/ik.