Fourier Series

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|Sx| 12a 0 ∑n1≤ 12a 0 ∑n1≤ 1 a 2 0 ∑n1 1 a 2 0 4M∑n1a n cosnx b n sinnx|a n cosnx| |b n sinnx||a n | |b n |1n 2 |Sx| 12a 0 2M 3 2 as ∑n11n 2 26 .Thus the series converges and hence the required result. It does not determine the value towhich the series converges, since this must depend upon the value of x.1.3.2 The Dirichlet ConditionsFor functions that are piecewise continuous or with discontinuous derivatives, the simplestset of conditions is provided by the Dirichlet conditions.If fx is defined and bounded on the range −, and if fx satisfies the followingconditions• fx has only a finite number of maxima and minima,• fx has a finite number of finite discontinuities,• fx is periodic with period 2,then the Fourier seriesSx 1 2 a 0 ∑n1a n cosnx b n sinnx ,with Fourier coefficients as defined previously, is convergent and converges to the sum12 fx 0 fx − 0. 18
Figure 1.3.1. Convergence of FS.Thus the series converges to the value of the function at points of continuity and to themidpoint between the two limits at a point of discontinuity. As it depends upon the limitsat a point of discontinuity, there is no requirement that the function is itself defined at thispoint.There are conditions that permit fx to be less well-behaved, while retaining convergenceof the series, but these are not needed at this point since the Dirichlet conditions areapplicable to most functions of interest. It is to be stressed that the Dirichlet conditions donot just provide a measure of convergence; they also provide a filtering mechanism todetermine whether or not a Fourier series representation can be obtained. For this reason,the function should be checked against the Dirichlet conditions before its Fourier series issought.Example 1Can the function defined by fx 1/4 − x 2 in −, be written as a Fourier series?No: It has infinite discontinuities at x 2 and so fails to satisfy the Dirichlet Conditions.Example 2Can the function defined by fx sin 1 in −, be written as a Fourier series?x−1No: The function has an infinite number of maxima and minima in the neighbourhood of19
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 14 and 15: than in a formula. Periodicity can
Page 16 and 17: 32.521.510.5000.5 1 1.5 2 2.5 3xFig
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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