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Unformatted text preview: the Fourier expansion of f restricts to an expansion of f of
the form (3.11) which involves only cosine functions. We call (3.11) the Fourier
cosine series of f .
To generate formulae for the coeﬃcients in the Fourier sine and cosine series
of f , we begin by deﬁning a slightly diﬀerent inner product space than the one 72 considered in the preceding section. This time, we let V be the set of piecewise
smooth functions f : [0, L] → R and let
V0 = {f ∈ V : f (0) = 0 = f (L)},
a linear subspace of V . We deﬁne an inner product ·, · on V by means of the
formula
2L
f, g =
f (t)g (t)dt.
L0
This restricts to an inner product on V0 .
Let’s consider now the Fourier sine series. We have seen that any element
of V0 can be represented as a superpostion of the sine functions
sin(πt/L), sin(2πt/L), ... , sin(nπt/L), ... . We claim that these sine functions form an orthonormal basis for V0 with respect
to the inner product we have deﬁned. Recall the trigonometric formulae that
we used in §3.1:
cos((n + m)πt/L) = cos(nπt/L) cos(mπt/L) − sin(nπt/L) sin(mπt/L),
co...
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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