B find the fourier cosine series of the same function

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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.

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