14 an 2 2 t cos ntdt t cos ntdt 0 2 this

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Unformatted text preview: is odd that all of the an ’s are zero. On the interval [0, L], f restricts ˆ fact that f ˆ to f and the Fourier expansion of f restricts to an expansion of f of the form (3.10) which involves only sine functions. We call (3.10) the Fourier sine series of f . A similar argument can be used to express a piecewise smooth function f : [0, L] → R into a superposition of cosine functions, f (t) = a0 + a1 cos(πt/L) + a2 cos(2πt/L) + . . . + an cos(nπt/L) + . . . . (3.11) 2 ˜ To obtain this expression, we first extend f to an even function f : [−L, L] → R by setting f (t), for t ∈ [0, L], ˜ f (t) = f (−t), for t ∈ [−L, 0], ˆ then to a function f : R → R which is periodic of period 2L, by requiring that ˆ ˆ f (t + 2L) = f (t), for all t ∈ R R. This time the extended function lies in the linear subspace Weven . It follows ˆ from the theorem in Section 3.1 that f possesses a Fourier series expansion, and ˆ from the fact that f is even that all of the bn ’s are zero. On the interval [0, L], ˆ ˆ f restricts to f and...
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