F t 315 in this case l and according to our

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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 ). 2 e. f (t) = 1 (et − e−t ). 2 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.

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