Fourier Series

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than in a formula. Periodicity can play strange tricks!Figure 1.2.1. The periodic function fx x.It is clear from figure 1.2.1 that the periodic function possesses discontinuities atx ,3,5,...To obtain the <strong>Fourier</strong> coefficients, use (1.2.2) and (1.2.4):a 0 1 −xdx 1 2x 2 2 − 0a m 1 x cosmx dx −1 1m cosmx 2 −1 xm sinmx −−1 − 1m 2 −1m − −1 m 01msinmx dxb m 1 −x sinmx dx 1 − x m cosmx −1 −1mcosmx dx 1 − m −1m −− m −1m 2 m −1m1where the result cosm −1 m has been used on a number of occasions in the derivation.Thus the final result isx~2 ∑n1−1n1nsinnx 2 sinx − 1 2 sin2x 1 3 sin3x − 1 4sin4x ... .The fact that the resulting series contains only sine terms is not surprising, since fx x isan odd function and so the series should only be composed of terms with the same property.14
We can check the progress of the approximation by plotting the function fx x andlooking at successive approximations; these are shown in figures 1.2.2(a)-(c) for the firstterm, first two terms and first three terms respectively.32.521.510.5000.511.522.53xFigure 1.2.2(a). The first term approximation.32.521.510.5000.511.522.53xFigure 1.2.2(b). The two-term approximation.15
Page 2 and 3: f 3 x 1 0 ≤ x 1 3 1 x ≤ 2Usi
Page 4 and 5: I 1 0−afx gx dx a0f−u g−u d
Page 6: 1.1.2(c) respectively.10.80.60.40.2
Page 10 and 11: dfdx limx→0fx x − fxx,which i
Page 12 and 13: −fxsinmx dx ∑ a n −n1cosnxsin
Page 16 and 17: 32.521.510.5000.5 1 1.5 2 2.5 3xFig
Page 18 and 19: |Sx| 12a 0 ∑n1≤ 12a 0 ∑n1
Page 20 and 21: x 1 and fails to satisfy the Diric
Page 22 and 23: 1.4 Half-range Fourier Series1.4.1
Page 24 and 25: f E x 1 fx f−x2 1 2 x − x
Page 26 and 27: defining our new function Fx to beF
Page 28 and 29: Figure 1.4.4. Extension of half-ran
Page 30 and 31: Figure 1.5.1. The periodic function
Page 32 and 33: 1.6 Integration and Differentiation
Page 34 and 35: If fx is continuous in the range
Page 36 and 37: c n 1 2 a n − ib n n 0 1 2 a 0
Page 38 and 39: c n 120e i1−nx dx in − 1 −1n
Page 40 and 41: 1.8 Evaluation and ImplementationMo
Page 42 and 43: The value of the series at these po
Page 44 and 45: ND N x 1 2∑n1Ncosnx ∑ e inx
Page 46 and 47: 1.9 Orthogonal Functions and Fourie
Page 48 and 49: unfortunately it is the same standa
Page 50 and 51: n 1,...N. This is true for all N a
Page 52 and 53: solutions of singular second order
Page 54: e x ~ 1 2 e − 1 e 3x e 5 4 e

Fourier Series

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