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# Prove the orthogonality relations for sine and cosine using complex exponentials

1. Prove the integral formula,

for integers and .

2. Prove the following orthogonality relations of sine and cosine using the relation in part (a), where and are integers with .

1. Proof. First, if then we have

If then we have

2. 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,

### One comment

1. S says:

I think there is a typo (or a mistake) in part a), because \int{exp(ikx)dx} = exp(ikx)/ik.