# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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f (x + T) = f (x) (2)

for every value of x. An example of a periodic function is shown in Figure 10.2.1. It follows immediately from the definition that if T is a period of f, then 2T is also a period, and so indeed is any integral multiple of T.

The smallest value of T for which Eq. (2) holds is called the fundamental period of f. In this connection it should be noted that a constant may be thought of as a periodic function with an arbitrary period, but no fundamental period.

1

--T--- x

FIGURE 10.2.1 A periodic function.

10.2 Fourier Series

549

If f and g are any two periodic functions with common period T, then their product fg and any linear combination c1 f + c2g are also periodic with period T. To prove the latter statement, let F(x) = c1 f (x) + c2g(x); then for any x

F (x + T) = Cj f (x + T) + C2g(x + T) = Cj f (x) + C2g(x) = F (x). (3)

Moreover, it can be shown that the sum of any finite number, or even the sum of a convergent infinite series, of functions of period T is also periodic with period T.

In particular, the functions sin(mnx/L) and cos(mnx/L), m = 1, 2, 3,..., are periodic with fundamental period T = 2L/m. To see this, recall that sin x and cos x have fundamental period 2n, and that sin ax and cos ax have fundamental period 2n/a. If we choose a = mn/L, then the period T of sin(mnx/L) and cos(mnx/L) is given by T = 2nL/mn = 2L/m.

Note also that, since every positive integral multiple of a period is also a period, each of the functions sin(m nx /L) and cos (m nx/L) has the common period 2L.

Orthogonality of the Sine and Cosine Functions. To describe a second essential property of the functions sin(mnx/L) and cos(mnx/L) we generalize the concept of orthogonality of vectors (see Section 7.2). The standard inner product (u, v) of two real-valued functions u and v on the interval a < x < â is defined by

f â

(u, v) = I u(x)v(x) dx. (4)

J a

The functions u and v are said to be orthogonal on a < x < â if their inner product is zero, that is, if

, â

u(x)v(x) dx = 0. (5)

a

a

A set of functions is said to be mutually orthogonal if each distinct pair of functions in the set is orthogonal.

The functions sin(mnx/L) and cos(mnx/L), m = 1, 2,..., form a mutually orthogonal set of functions on the interval —L < x < L .In fact, they satisfy the following orthogonality relations:

fL m n x nn x [ 0, m = n,

cos------cos-------dx = ³ ’ ' (6)

J—L L L \L, m = n;

L~ L- L

[L . mnx . nnx [0, m = n,

sin------sin-------dx = { ’ ' (8)

-L L L L , m = n.

These results can be obtained by direct integration. For example, to derive Eq. (8), note that

r'L mnx . nnx

cos------------sin------------dx = 0, all m, n; (7)

f'i . mnx . nnx 1 fL

sin-sin--dx = - I

—LLL 2 J—L

(m — n)n x (m + n)n x

cos -------Y,---------cos --------^--------

dx

L

1 L f sin[(m — n)nx/L] sin[(m + n)nx/L]1

2 n [ m — n m + n J

-L

= 0,

550

Chapter 10. Partial Differential Equations and Fourier Series

as long as m + n and m — n are not zero. Since m and n are positive, m + n = 0. On the other hand, if m — n = 0, then m = n, and the integral must be evaluated in a different way. In this case

*L r L

/L . mnx . nnx fL / . mnx\2

sin----sin-----dx = I I sin------I dx

L L L J—L^ L '

=11

1 f sin(2m n x / L)

- \x -

dx

L

—L

2 \ 2m n/L

= L.

This establishes Eq. (8); Eqs. (6) and (7) can be verified by similar computations.

The Euler-Fourier Formulas. Now let us suppose that a series of the form (1) converges, and let us call its sum f (x):

a0 / mnx mnx \

f (x) = f + E \m cos — + bm sin —) . (9)

m=1

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