# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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Also of importance are the following two integral properties of even and odd functions:

4. If f is an even function, then

f f (x) dx = 2 f f (x) dx. (5)

J-L J0

5These statements may need to be modified if either function vanishes identically.

566

Chapter 10. Partial Differential Equations and Fourier Series

i:

5. If f is an odd function, then

» L

f (x) dx = 0. (6)

l-L

These properties are intuitively clear from the interpretation of an integral in terms of area under a curve, and also follow immediately from the definitions. For example, if f is even, then

/L /»0 cL

f (x) dx = I f (x) dx + I f (x) dx.

L -L 0

Letting x = 5 in the first term on the right side, and using Eq. (1), we obtain

/L /»0 cL f*L

f (x) dx = I f (s) ds + I f (x) dx = 2 1 f (x) dx.

l Jl Jo Jo

The proof of the corresponding property for odd functions is similar.

Even and odd functions are particularly important in applications of Fourier se-

ries since their Fourier series have special forms, which occur frequently in physical problems.

Cosine Series. Suppose that f and f' are piecewise continuous on L < x < L, and that f is an even periodic function with period 2L. Then it follows from properties 1 and 3 that f (x) cos(nnx/L) is even and f (x) sin(nnx/L) is odd. As a consequence of Eqs. (5) and (6), the Fourier coefficients of f are then given by

2 fL nnx

an = I f (x) cos dx, n = 0, 1, 2,...;

L Jo l (7)

= 0, n = 1, 2,....

Thus f has the Fourier series

a0 nn x

f (x) = if an c°s_T

n = 1

In other words, the Fourier series of any even function consists only of the even trigonometric functions cos(nnx/L) and the constant term; it is natural to call such a series a Fourier cosine series. From a computational point of view, observe that only the coefficients an, for n = 0, 1, 2,, need to be calculated from the integral formula (7). Each of the bn, for n = 1, 2,..., is automatically zero for any even function, and so does not need to be calculated by integration.

Sine Series. Suppose that f and f' are piecewise continuous on L < x < L, and that f is an odd periodic function of period 2L. Then it follows from properties 2 and 3 that f (x) cos(nnx/L) is odd and f (x) sin(nnx/L) is even. In this case the Fourier coefficients of f are

a = 0, n = 0, 1, 2,...,

2 fL nn x (8)

bn If (x) sin - dx, n = 1, 2,...,

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10.4 Even and Odd Functions

567

EXAMPLE

1

and the Fourier series for f is of the form

CO

, . nnx

f (x) = bnsin

n=1

Thus the Fourier series for any odd function consists only of the odd trigonometric functions sin(nnx/L); such a series is called a Fourier sine series. Again observe that only half of the coefficients need to be calculated by integration, since each an, for n = 0, 1, 2,, is zero for any odd function.

Let f (x) = x, L < x < L, and let f (L) = f (L) = 0. Let f be defined elsewhere so that it is periodic of period 2L (see Figure 10.4.2). The function defined in this manner is known as a sawtooth wave. Find the Fourier series for this function.

Since f is an odd function, its Fourier coefficients are, according to Eq. (8),

n = 0, 1, 2,...;

an = 0

2 fL nnx

b = I x sin ax

n L J0 L

2

2 / L I . nnx nnx nnx

= Lynn) (sin ~L ~Tcos ~L~

= (1)n+1, nn

n = 1, 2,

Hence the Fourier series for f, the sawtooth wave, is

2L

f (x' =

(1)

n+1

nn x

sin

n1

L

(9)

Observe that the periodic function f is discontinuous at the points ±L, ±3 L,..., as shown in Figure 10.4.2. At these points the series (9) converges to the mean value of the left and right limits, namely, zero. The partial sum of the series (9) for n = 9 is shown in Figure 10.4.3. The Gibbs phenomenon (mentioned in Section 10.3) again occurs near the points of discontinuity.

L

n

FIGURE 10.4.2 Sawtooth wave.

568

Chapter 10. Partial Differential Equations and Fourier Series

/ y- L 1/

/-2L -L / L / 2L x

n = 9^ / /

- L - f

FIGURE 10.4.3 A partial sum in the Fourier series, Eq. (9), for the sawtooth wave.

Note that in this example f (-L) = f (L) = 0, as well as f (0) = 0. This is required if the function f is to be both odd and periodic with period 2L. When we speak of constructing a sine series for a function defined on 0 < x < L, it is understood that, if necessary, we must first redefine the function to be zero at x = 0 and x = L.

It is worthwhile to observe that the triangular wave function (Example 1 of Section

10.2) and the sawtooth wave function just considered are identical on the interval 0 < x < L. Therefore, their Fourier series converge to the same function, f (x) = x, on this interval. Thus, if it is required to represent the function f (x) = x on0 < x < L by a Fourier series, it is possible to do this by either a cosine series or a sine series. In the former case f is extended as an even function into the interval L < x < 0 and elsewhere periodically (the triangular wave). In the latter case f is extended into L < x < 0 as an odd function, and elsewhere periodically (the sawtooth wave). If f is extended in any other way, the resulting Fourier series will still converge to x in 0 < x < L but will involve both sine and cosine terms.

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