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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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 Winter '14
 Equations

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