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Unformatted text preview: ermine which of the following functions are even, which are odd, and
which are neither even nor odd:
a. f (t) = t3 + 3t.
b. f (t) = t2 + t.
c. f (t) = et .
d. f (t) = 1 (et + e−t ).
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e. f (t) = 1 (et − e−t ).
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f. f (t) = J0 (t), the Bessel function of the ﬁrst kind. 71 3.3 Fourier sine and cosine series Let f : [0, L] → R be a piecewise smooth function which vanishes at 0 and L.
We claim that we can express f (t) as the superposition of sine functions,
f (t) = b1 sin(πt/L) + b2 sin(2πt/L) + . . . + bn sin(nπt/L) + . . . . (3.10) We could prove this using the theory of even and odd functions. Indeed, we can
˜
extend f to an odd function f : [−L, L] → R by setting
˜
f (t) = f (t),
for t ∈ [0, L],
−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 . The extended function lies in the linear subspace Wodd . It follows from the
ˆ
theorem in Section 3.1 that f possesses a Fourier series expansion, and from the
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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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