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In section 3.5.4 we showed that the ordinary Fourier series of an even periodic tion contains only cosine terms and that the Fourier series of an odd periodic func-tion contains only sine terms. For the standard funcfunc-tions we have seen that the pe-riodic block function and the pepe-riodic triangle function, which are even, do indeed contain cosine terms only and that the sawtooth function, which is odd, contains sine terms only. Sometimes it is desirable to obtain for an arbitrary function on the interval(0, T ) a Fourier series containing only sine terms or containing only cosine terms. Such series are called Fourier sine series and Fourier cosine series. In order Fourier sine series

Fourier cosine series to find a Fourier cosine series for a function defined on the interval(0, T ), we extend the function to an even function on the interval(−T, T ) by defining f (−t) = f (t) for−T < t < 0 and subsequently extending the function periodically with period

2T . The function thus created is now an even function and its ordinary Fourier series will contain only cosine terms, while the function is equal to the original function on the interval(0, T ).

In a similar way one can construct a Fourier sine series for a function by extending the function defined on the interval(0, T ) to an odd function on the interval (−T, T ) and subsequently extending it periodically with period 2T . Such an odd function will have an ordinary Fourier series containing only sine terms.

Determining a Fourier sine series or a Fourier cosine series in the way described Forced series development

above is sometimes called a forced series development.

Let the function f(t) be given by f (t) = t²on the interval(0, 1). We wish to obtain EXAMPLE

a Fourier sine series for this function. We then first extend it to an odd function on the interval(−1, 1) and subsequently extend it periodically with period 2. The function and its odd and periodic extension are drawn in figure 3.13. The ordinary

1

The function f(t) = t²on the interval(0, 1) and its odd and periodic extension.

Fourier coefficients of the function thus created can be calculated using (3.5) and (3.6). Since the function is odd, all coefficients a_nwill equal 0. For b_nwe have

b_n = 2

Applying integration by parts twice, it follows that bn = −2 In this final example we will show that one can even obtain a Fourier cosine series EXAMPLE

for the sine function on the interval(0, π). To this end we first extend sin t to an even function on the interval(−π, π) and then extend it periodically with period 2π; see figure 3.14. The ordinary Fourier coefficients of the function thus created can be calculated using (3.5) and (3.6). Since the function is even, all coefficients

π –π

–2π 2π

FIGURE 3.14

The even and periodic extension of the function f(t) = sin t on the interval (0, π).

bnwill be 0. For anone has

Applying the trigonometric formula sin t cos nt= (sin(1+n)t +sin(1−n)t)/2 then gives for anwith n= 1:

Determine the Fourier sine series and the Fourier cosine series on(0, 4) for the 3.19

function f(t) given for 0 < t < 4 by

f(t) =

 t for 0< t ≤ 2, 2 for 2< t < 4.

Determine a Fourier sine series of cos t on the interval(0, π).

3.20

Determine a Fourier sine series and a Fourier cosine series of the function f(t) = 3.21

t(t − 4) on the interval (0, 4).

Determine a Fourier sine series of the function f(t) defined on the interval (0, T/2) 3.22

by f(t) = 1/2 for 0 ≤ t < T/2.

S U M M A R Y

Trigonometric polynomials and series are, respectively, finite and infinite linear combinations of the functions cos nω0t and sin nω0t with n ∈ N. They are all periodic with period T = 2π/ω0. When a trigonometric polynomial f(t) is given, the coefficients in the linear combination can be calculated using the formulas an = 2

T

 _T/2

−T/2f(t) cos nω0t dt for n= 0, 1, 2, . . ., bn = 2

T

 _T/2

−T/2f(t) sin nω0t dt for n= 1, 2, . . ..

These formulas can be applied to any arbitrary periodic function, provided that the integrals exist. The numbers a_nand b_nare called the Fourier coefficients of the func-tion f(t). Using these coefficients one can then form a Fourier series of the func-tion f(t):

a₀ 2 +^∞

n=1

(ancos nω0t+ bnsin nω0t).

Instead of a Fourier series with functions cos nω0t and sin nω0t one can also obtain a complex Fourier series with the time-harmonic functions e^{i nω0t}. The complex Fourier coefficients can then be calculated using

cn= 1 T

 _T/2

−T/2 f(t)e^−inω0tdt for n∈ Z, while the complex Fourier series has the form

∞ n=−∞

c_ne^{i nω0t}.

These two Fourier series can immediately be converted into each other and, depend-ing on the application, one may chose either one of these forms.

The sequence of Fourier coefficients(cn) is called the spectrum of the func-tion. This is usually split into the amplitude spectrum| cn| and the phase spectrum arg(cn).

For some standard functions the Fourier coefficients have been determined. More-over, a number of properties were derived making it possible to find the Fourier coefficients for far more functions and in a much simpler way than by a direct calculation.

Even functions have Fourier series containing cosine terms only. Series like this are called Fourier cosine series. Odd functions have Fourier sine series. When desired, one can extend a function, given on a certain interval, in an even or an odd way, so that they can be forced into a Fourier cosine or a Fourier sine series.

S E L F T E S T

The function f(t) is periodic with period 10 and is drawn on the interval (−5, 5) in 3.23

figure 3.15. Determine the ordinary and complex Fourier coefficients of f . Show that when for a real function f the complex Fourier coefficients are real, f 3.24

has to be even, and when the complex Fourier coefficients are purely imaginary, f has to be odd.

–1 1 3 –3

–5 0

2

5 t

f(t)

FIGURE 3.15

The periodic function f(t) from exercise 3.23.

Determine the Fourier series of the periodic function f(t) with period T , when for 3.25

−T/2 < t < T/2 the function f (t) is given by f(t) =

 0 for−T/2 < t < 0, sinω0t for 0< t < T/2.

Calculate and sketch the amplitude and phase spectrum of the periodic function 3.26

f(t), when f (t) has period 2π and is given for −π < t < π by f(t) =

 0 for−π < t < 0, t for 0≤ t < π.

Consider the function f(t) defined by:

3.27

f(t) =







 2b

at for 0< t < a 2, 2b

a(a − t) for a

2 ≤ t < a.

a Sketch the graph of f(t), of its odd and of its even periodic extension.

b Give a development of f(t) on (0, a) as a Fourier cosine series and also as a Fourier sine series.

The fundamental theorem of Fourier series Introduction 86

4.1 Bessel’s inequality and Riemann–Lebesgue lemma 86 4.2 The fundamental theorem 89

4.3 Further properties of Fourier series 95 4.3.1 Product and convolution 96

4.3.2 Parseval’s identity 99 4.3.3 Integration 99 4.3.4 Differentiation 101

4.4 The sine integral and Gibbs’ phenomenon 105 4.4.1 The sine integral 106

4.4.2^∗ Gibbs’ phenomenon 107 Summary 109

Selftest 110

The fundamental theorem of Fourier