Page 41 - Essentials of applied mathematics for scientists and engineers

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CHAPTER 3

Orthogonal Sets of Functions

In this chapter we elaborate on the concepts of orthogonality and Fourier series. We begin
with the familiar concept of orthogonality of vectors. We then extend the idea to orthogonality
of functions and the use of this idea to represent general functions as Fourier series—series of
orthogonal functions.
Next we show that solutions of a fairly general linear ordinary differential equation—the
Sturm–Liouville equation—are orthogonal functions. Several examples are given.

3.1 VECTORS
We begin our study of orthogonality with the familiar topic of orthogonal vectors. Suppose u(1),
u(2), and u(3) are the three rectangular component vectors in an ordinary three-dimensional
space. The norm of the vector (its length) ||u|| is

2 1/2
2
2
||u|| = [u(1) + u(2) + u(3) ] (3.1)
If ||u|| = 1, u is said to be normalized. If ||u|| = 0, u(r) = 0for each r and u is the zero vector.
A linear combination of two vectors u 1 and u 2 is

u = c 1 u 1 + c 2 u 2 , (3.2)

The scalar or inner product of the two vectors u 1 and u 2 is defined as
3

(u 1 , u 2 ) = u 1 (r)u 2 (r) = u 1 u 2 cos θ (3.3)
r=1

3.1.1 Orthogonality of Vectors
If neither u 1 nor u 2 is the zero vector and if

(u 1 , u 2 ) = 0 (3.4)
then θ = π/2 and the vectors are orthogonal. The norm of a vector u is

||u|| = (u, u) 1/2 (3.5)