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2. The sum (difference) of two odd functions is odd; the product (quotient) of two odd functions is even.
3. The sum (difference) of an odd function and an even function is neither even nor odd; the product (quotient) of two such functions is odd.5
The proofs of all these assertions are simple and follow directly from the definitions. For example, if both f1 and f2 are odd, and if g(x) = f1(x) + f2(x), then
g(-x) = f 1 ( x) + f2( x) = -f1(x) - f2(x)
= - fx) + f2(x)] = -g(x), (3)
so f1 + f2 is an odd function also. Similarly, if h(x) = f1(x) f2(x), then
h(-x) = f1(-x )f2(-x) = [- f1(x)][- f2(x)] = f1(x )f2(x) = h(x), (4)
so that f1 f2 is even.
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 j f (x) dx. (5)
5These statements may need to be modified if either function vanishes identically.
Chapter 10. Partial Differential Equations and Fourier Series
5. If f is an odd function, then
f (x) dx = 0. (6)
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 fL
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 fL fL
f (x) dx = I f (s) ds + I f (x) dx = 2 I f (x) dx.
l Jl J0 J0
The proof of the corresponding property for odd functions is similar.
Even and odd functions are particularly important in applications of Fourier series 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 J0 L (7)
b = 0, n = 1, 2,....
Thus f has the Fourier series
a0 nn x
f (x) = if +E an cos.
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 nnx (8)
bn = I f (x) sin - dx, n = 1, 2,...,
L J0 L
10.4 Even and Odd Functions
and the Fourier series for f is of the form
, . nnx
f (x ) = ? bn si^-^ 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.