Let us use formula 321 to nd the complex fourier

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Unformatted text preview: ctions 1 √, 2 cos(πt/L), cos(2πt/L), ... , cos(nπt/L), ... form an orthonormal basis for V . Thus we can evaluate the coefficients of the Fourier cosine series of a function f ∈ V by projecting f onto each element of this orthonormal basis. We will leave it to the reader to carry this out in detail, and simply remark that when the dust settles, one obtains the following formula for the coefficient an in the Fourier cosine series: 2 L an = L f (t) cos(nπt/L)dt. (3.14) 0 Example. First let us use (3.13) to find the Fourier sine series of t, for 0 ≤ t ≤ π/2, π − t, for π/2 ≤ t ≤ π . f (t) = (3.15) In this case, L = π , and according to our formula, bn = 2 π π /2 π (π − t) sin ntdt . t sin ntdt + 0 π/2 We use integration by parts to obtain π /2 t sin ntdt = 0 = −t cos nt n π /2 π /2 + 0 0 1 cos ntdt n −π cos(nπ/2) −π cos(nπ/2) sin(nπ/2) 1 π/2 , + 2 [sin nt]|0 = + 2n n 2n n2 while π (π − t) sin ntdt = π/2 = −(π − t) cos(nt) n π π − π/2 π/2 1 cos ntdt n π cos(nπ/2) π c...
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