Fourier Series

More documents

Recommendations

Info

1.4 Half-range Fourier Series1.4.1 Fourier series for even functionsWe already know that an even function is defined byfx f−xand this property can be utilised in determining the Fourier coefficients. The generalcosine coefficient a n is given bya n 1 −fxcosnx dx 01 −fxcosnx dx 1 0fxcosnx dx 1 I 1 I 2 .ThenI 1 0−fxcosnx dx 0f−u cosn−u d−u where x −u f−ucosn−u du on simplification.0But f−u fu and cosn−u cosnu as both are even functions, thusI 1 fucosnu du I2 ,0 a n 2 fxcosnx dx0The general sine coefficient b n is given byb n 1 −fxsinnx dx .The integrand is the product of an odd and an even function, with the integration over aninterval that possesses x 0 as its midpoint. Thus our general result for integrals of thistype holds and b n 0. If the function is identified as being even then the forms above canbe used immediately and do not need to be proved on a regular basis.1.4.2 Fourier series for odd functionsSimilar arguments apply when the function is an odd function and satisfiesUsing arguments similar to those abovefx − f−x .22
a n 0, b n 2 0fxsinnx dx .1.4.3 Decomposition into even and odd functions.It has already been shown that a general function fx on an arbitrary interval −a,a can bewritten as the sum of an even and an odd function by the decompositionfx 1 fx f−x 2 1fx − f−x2 f E x f O x Even function Odd function .In the context of a Fourier series, the sums of cosine and sine terms must correspond toeven and odd functions respectively as the individual terms are themselves either even orodd.Suppose that a function fx is defined on −,, is there any benefit to be gained bydecomposing it into an even and an odd component before the Fourier series is sought? Inmost cases the answer is negative, since only the Fourier series is required and the even andodd components are associated with the cosine and sine terms. However, it mayoccasionally be easier to determine the series by considering the even and odd componentsseparately, with the advantage of requiring only the half-range formulae.ExampleConsider the function defined byfx −x − x 0 x − 0 x and decompose this into the sum of even and odd functions.It is not immediately obvious from the schematic in figure 1.4.1 how the even and oddcharacteristics of the function will manifest themselves but the presence of the |x|component suggests that a decomposition should not be difficult to obtain. Follow thestandard procedures, ensuring that the implementation is made separately for x 0andx 0. With the suffices E and O denoting the even and odd components respectively, theeven component is23
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 18 and 19: |Sx| 12a 0 ∑n1≤ 12a 0 ∑n1
Page 20 and 21: x 1 and fails to satisfy the Diric
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

Create successful ePaper yourself

Delete template?

Save as template?