Fourier Series

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ND N x 1 2∑n1Ncosnx ∑ e inx . (1.8.3)n−NIt is straightforward to show that the second form is the same as the first, since D N x is aneven function. Note that although D N x is even, its contribution via the integrand of(1.8.2) is not that of an even function due to the delay associated with the evaluation ofx − t.From (1.7.9), the general convolution of two periodic functions satisfiesand so (1.8.2) gives fx − t gt dt − ft gx − t dt−S N x 12−ft D N x − tdt 12−D N t fx − tdt , (1.8.4)which provides an alternative representation in which the contribution from D N x is that ofan even function.The series for D N x is finite for finite N and can be summed using the same method asemployed when summing the integrand in the Gibb’s Phenomenon example in the previoussection. This givesD N x sinN 1 2 xsin x 2.By employing the second of the two integral representations for S N x in (1.8.4), togetherwith the function fx 1, it is straightforward to show that0DN t dt ,for all N and which can also be seen from the series definition in (1.8.3).Substituting the summed form for D N x into the first expression for S N x in (1.8.4), thepartial sum becomesS N x 12−ft sinN 1 x − t2sinx − t/2dtand this identifies an important practical point. If the series Sx is to be obtained via asequence of partial sums, with S N x given by the form above, then it is unlikely that theexpression can be evaluated analytically and numerical methods would be required. As Nincreases the integrand oscillates increasingly due to the sinN ½x − t term and this44
can cause numerical difficulties if N is large, so employment of the Dirichlet Kernel D N xto evaluate Sx may appear attractive in principle but may not be feasible in practice.1.8.4 MiscellaneousThe Dirichlet conditions cover most commonly used functions but are unnecessarilyrestrictive in a general theory. This is a major topic and best considered by reference to therelevant references.Example: FS can be obtained and are convergent for functions that are absolutelyintegrable, such as logx, log2sinx/2 and log2cosx/2.45
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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 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 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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