The facts that make Fourier analysis work are that for different speeds v and w
/T | (*1) (1/T)| cos(vt) cos(wt) dt ---> 0 | /0as T --> infty
and similarly for
(*2) cosvt sinwt,
(*3) sinvt sinwt, and
(*4) cosvt sinvt; whereas
/T | 2 (*5) (1/T)| cos (vt) dt ---> 1/2 | /0as T --> infty
and the same for
/T | 2 (*6) (1/T)| sin (vt) dt. | /0
Each of these facts follows from a trigonometric identity.
For (*1), the identity is
an easy consequence of the addition formula. Substituting this identity in the integral for (*1) gives
/T | 1 1 2| cos(vt) cos(wt) dt = --- sin(vT+wT) + --- sin(vT-wT) | v+w v-w /0which as a function of T is bounded in absolute value by the sum of the absolute values of (1/(v+w)) and (1/(v-w)) (note here how the hypothesis "v,w different" is necessary!); so then the integral is divided by T and T --> infty the limit of the quotient is zero.
For (*2) the relevant identity is
For (*3) it is
The limit for (*4) is established by noting that
1 d 2 cosvt sin vt = - -- sin vt. 2 dt
On the other hand, to establish (*5) it is sufficient to use the identity
/T | T 1 | cosvt cosvt dt = - + -- sin(2vT). | 2 2v /0Since the second term is bounded in absolute value (by 1/2v), dividing by T and letting T --> infty gives the limit 1/2. In the same way, (*6) follows from the identity
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