Fourier Series

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f E x 1 fx f−x2 1 2 x − x x − 2 1 2 −x −x − −x − 20 x − x 0which can also be written asf E x |x| − 2 .Figure 1.4.1. Schematic of function.The resulting odd component isf O x 1 fx − f−x2 1 2 x − − x − 2 1 2 −x − −x − 20 x − x 0.Both functions are shown in figure 1.4.2 below, with the even and odd functions depictedas a solid and broken line respectively.24
Figure 1.4.2. Decomposition into even and odd componentsThe resulting components of the FS are given bya n 2 0fE xcosnx dx, b n 2 0fO x sinnx dx ,which are straightforward to evaluate.1.4.4 Half-range SeriesSuppose that the function is defined on 0,. Any function defined on just this domainwill not be 2-periodic and not amenable to existing techniques, as we do not yet have thetechniques available for a domain of arbitrary length.However we can easily construct a new function Fx that is 2-periodic, defined upon−, and coincides with fx on 0,. There are an infinite number of possibilities andin the first instance we employ either a Half-range Sine Series or a Half-range CosineSeries, as these are the simplest such extensions and can have evenness or oddness bychoice. Each of these representations only represents the original function in 0,, hence"half-range". This is easiest seen by an example.Example 1Suppose that fx cosx in 0,. We extend this to −, as a half-range sine series by25
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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 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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