COMPLETE SETS OF ORTHOGONAL FUNCTIONS

6. COMPLETE SETS OF ORTHOGONAL FUNCTIONS

Orthogonal functions Two vectors A and B are orthogonal (perpendicular) if their scalar product is zero, that is, if

A i B i = 0.

[See Chapter 3, equations (4.12) and (10.3).] Recall from Chapter 3, Section 14, that we can think of functions as elements of a vector space. Then by analogy with (6.1) we say that two functions A(x) and B(x) are orthogonal on (a, b) if

A(x)B(x) dx = 0.

If the functions A(x) and B(x) are complex, the definition of orthogonality is [see Chapter 3, equation (14.3)]

A(x) and B(x) are orthogonal on (a, b) if

A ∗ (x)B(x) dx = 0,

where A ∗ (x) is the complex conjugate of A(x) (see Problem 1).

Since (6.3) is identical with (6.2) if A(x) and B(x) are real, we can take (6.3) as the general definition of orthogonality of A(x) and B(x) on (a, b).

If we have a whole set of functions A n (x) where n = 1, 2, 3, · · · , and

A ∗ n (x)A m (x) dx =

we call the functions A n (x) a set of orthogonal functions. We have already used such sets of functions in Fourier series. Recall that [Chapter 7, equation (5.2)]

Thus sin nx is a set of orthogonal functions on (−π, π), or in fact on any other interval of length 2π. Similarly, the functions cos nx are orthogonal on (−π, π).

576 Series Solutions of Differential Equations Chapter 12

Also the whole set consisting of sin nx and cos nx is a set of orthogonal functions on (−π, π) since

sin nx cos mx dx = 0 for any n and m.

We have used complex functions also, namely the set e inx . For this set the orthog- onality property is given by (6.4), namely

e −inx e imx dx =

2π if m = n. Recall that sin nx and cos nx (or e inx ) were the functions used in a Fourier se-

ries expansion on (−π, π). You should now realize that it was the orthogonality property that we used in getting the coefficients. When we multiplied the equation

f (x) = ∞ m=−∞ c m e imx by e −inx and integrated, the integrals of all the terms in the series except the c n term were zero by the orthogonality property (6.6). There are many other sets of orthogonal functions besides the trigonometric or exponential ones. Just as we used the sine-cosine or exponential set to expand a function in a Fourier series, so we can expand a function in a series using other sets of orthogonal functions. We shall show this for the functions P l (x) after we prove that they are orthogonal.

Complete sets There is another important point to consider when we want to expand a function in terms of a set of orthogonal functions. Again let us consider the vector analogy. We write vectors in terms of their components and the basis vectors i, j, k. In two dimensions we need only two basis vectors, say i and j. But if we tried to write three-dimensional vectors in terms of just i and j, there would be some vectors we could not represent; we say that (in three dimensions)

i and j are not a complete set of basis vectors. A simple way of expressing this (which generalizes to n dimensions) is to say that there is another vector (namely k) which is orthogonal to both i and j. Thus we define a set of orthogonal basis vectors as complete if there is no other vector orthogonal to all of them (in the space of the number of dimensions we are considering). By analogy, we define a set of orthogonal functions as complete on a given interval if there is no other function orthogonal to all of them on that interval. Now it is easy to see that there are some vectors in three dimensions which cannot be represented using only i and j . Similarly, there are functions which cannot be represented by a series using an incomplete set of orthogonal functions. We have discussed one example of this in Fourier series (Chapter 7, Section 11). If we are trying to represent a sound wave by a Fourier series, we must not leave out any of the harmonics; that is, the set of functions sin nx, cos nx on (−π, π) would not be complete if we left out some of the values of n. As another example, the set of functions sin nx is an orthogonal set on (−π, π). However, it is not complete; to have a complete set we must include also the functions cos nx, and you should recall that this is what we did in Fourier series. On the other hand, sin nx is a complete set on (0, π); we used this fact when we started with a function given on (0, π), defined it on (−π, 0) to make it odd, and then expanded it in a sine series. Similarly, cos nx is a complete set on (0, π). In this chapter, we are particularly interested in the fact (which we state without proof) that the Legendre polynomials are a complete set on (−1, 1).

Section 7 Orthogonality of the Legendre Polynomials 577

PROBLEMS, SECTION 6

R b 1. ∗ Show that if a A (x)B(x) dx = 0 [see (6.3)], then a A(x)B (x) dx = 0, and vice versa.

2. Show that the functions e inπx/l , n = 0, ±1, ±2, · · · , are a set of orthogonal functions on (−l, l).

3. Show that the functions x 2 and sin x are orthogonal on (−1, 1). Hint: See Chapter 7, Section 9.

4. Show that the functions f (x) and g(x) are orthogonal on (−a, a) if f(x) is even and g(x) is odd. (See Problem 3.)

5. R Evaluate 1 − 1 P 0 (x)P 2 (x) dx to show that these functions are orthogonal on (−1, 1). 6. Show in two ways that P l (x) and P l ′ (x) are orthogonal on (−1, 1). Hint: See Prob-

lem 4 and Problem 4.4. 7. Show that the set of functions sin nx is not a complete set on (−π, π) by trying to

expand the function f (x) = 1 on (−π, π) in terms of them. 8. Show that the functions cos (n + 1 2 )x, n = 0, 1, 2, · · · , are orthogonal on (0, π). Ex-

pand the function f (x) = 1 on (0, π) in terms of them. (Is it a complete set? See Chapter 7, end of Section 11.) R 9. 1 Show in two ways that

−1 P 2n+1 (x) dx = 0.