Fourier Series

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1.9 Orthogonal Functions and <strong>Fourier</strong> Theory1.9.1 Orthogonal and Orthonormal FunctionsIf a set of continuous functions n x,n 1,2,3,...., defined on the interval a,b, hasthe property thatabm x n x dx gm mn , (1.9.1)where g is some function of m, then the set is orthogonal.If the set satisfiesabm x n x dx mn , (1.9.2)i.e. gm ≡ 1, for all m, then the set is orthonormal.A set that is orthogonal but not orthonormal can be transformed into an orthonormal one bydividing each member of the set m x by gm . Clearly this procedure can only work ifgm ≠ 0 for all m.The set is complete if the conditions x ≡ 0.abn xx dx 0, n 1,2,3,....Note that gm cannot be zero for a non-zero m x. If it were zero, then (1.9.1) wouldgiveabm x m x dx ab|m x| 2 dx 0and as the integrand is positive over a finite domain, the only possibility for a continuousfunction is m x 0.ExampleIn the interval −L,L, show that the set of functions n x 1 Lsin nxL, n 1,2,3.....form an orthonormal set.To show this, we have to confirm that the n x satisfy (1.9.2) above.46
L−L m x n x dx L1L −L1 −sin nxLsin mxLdxz x/Lsinnz sinmz dz 1 . mn mnHence the required result but is the set complete? The answer to this question is no, sinceall of the m x are odd functions andL−L m xx dx 0for any even function x and completeness would require that x 0 to be the onlypossibility.Suppose that an arbitrary function fx is defined on a,b, how can this be represented as aseries of orthonormal functions? Alternatively, if we writefx ~ ∑ a n n x ,n1how do we determine the a n s? There are inevitably issues of convergence here, such aswhether or not the series converges and what it converges to; but these are ignored in thepresent discussion.Proceed as with FS, multiply both sides by m x for a fixed m, and integrate over theinterval,abfxm x dx ~∑n1∑n1a n abn x m x dxa n mn a mThusfx ~∑n1∑n1abfxn x dx n xf, n n x (1.9.3)where f, n is called the Inner Product of fx and n x on a,b Beware! Take carenot to confuse the inner product notation with that employed for an open interval;47
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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 44 and 45: ND N x 1 2∑n1Ncosnx ∑ e inx
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

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