Introduction to Fourier Series

MATH 467 Partial Differential Equations

J. Robert Buchanan

Department of Mathematics

Spring 2016

Objectives

In this lesson we will learn:

▶ the formal process for finding a Fourier series

representation of a function, ▶ the orthogonality of the trigonometric functions, ▶ the Euler-Fourier formulas for finding Fourier series

coefficients, ▶ properties of periodic functions, ▶ how to periodically extend a function, ▶ the properties of even and odd periodic extensions of

functions, and ▶ practice finding the Fourier series representations of

functions.

Informal Definition of a Fourier Series

The Fourier series expansion of a function f (x) is a representation of f (x) on an interval [−L, L] as the sum of sine and cosine functions of the form

\frac{a_{0}}{2} + ∑_{n=1}^{∞}(a_{n} cos \frac{nπx}{L} + b_{n} sin \frac{nπx}{L})

where a_{n} and b_{n} are constants.

Issues Raised by Fourier Series

▶ What functions f (x) can be written as a Fourier series? ▶ If f (x) can be represented as a Fourier Series, what are

the constants a_{n} and b_{n}? ▶ Will the Fourier series converge? ▶ Provided the Fourier series converges, does it converge to

f (x) at all points in the interval [−L, L]? ▶ Can Fourier series be differentiated and integrated?

Inner Product

Definition

If u(x) and v(x) are integrable on [a, b], the inner product of u and v on [a, b], denoted as ⟨u, v⟩, is defined as

⟨u, v⟩ = ∫_{a}^{b} u(x)v(x) dx.

Definition

The functions u and v are said to be orthogonal on [a, b] if

⟨u, v⟩ = ∫_{a}^{b} u(x)v(x) dx = 0.

A set S of integrable functions on [a, b] is said to be a mutually orthogonal set if each pair of distinct functions in the set is orthogonal.

Inner Product

Definition

If u(x) and v(x) are integrable on [a, b], the inner product of u and v on [a, b], denoted as ⟨u, v⟩, is defined as

⟨u, v⟩ = ∫_{a}^{b} u(x)v(x) dx.

Definition

The functions u and v are said to be orthogonal on [a, b] if

⟨u, v⟩ = ∫_{a}^{b} u(x)v(x) dx = 0.

A set S of integrable functions on [a, b] is said to be a mutually orthogonal set if each pair of distinct functions in the set is orthogonal.

Trigonometric System

Let S be the infinite set of functions \{1, cos \frac{πx}{L}, sin \frac{πx}{L}, cos \frac{2πx}{L} , sin \frac{2πx}{L} , ⋯, cos \frac{nπx}{L}, sin \frac{nπx}{L}, ⋯ \} . S is a mutually orthogonal set on [−L, L].

Product-to-Sum Formulas

cos α cos β = \frac{1}{2} (cos (α + β) + cos (α − β))
cos α sin β = \frac{1}{2} (sin(α + β) − sin(α − β))
sin α sin β = \frac{1}{2} (cos (α − β) − cos (α + β))

Justification of Orthogonality

∫_{−L}^{L}cos \frac{mπx}{L} cos \frac{nπx}{L} dx

= \frac{1}{2} ∫_{−L}^{L}[cos \frac{(m + n)πx}{L} + cos \frac{(m − n)πx}{L}] dx
= \{ \begin{matrix}\frac{L}{2π} [ \frac{1}{m + n} sin \frac{(m + n)πx}{L} + \frac{1}{m − n} sin \frac{(m − n)πx}{L}]_{−L}^{L} & if m ≠ n, \\ \frac{1}{2} [ \frac{L}{2mπ} sin \frac{2mπx}{L} + x]_{−L}^{L} & if m = n \end{matrix}
= \{ \begin{matrix}0 & if m ≠ n, \\ L & if m = n. \end{matrix}

The orthogonality of sin(mπx/L), sin(nπx/L), and cos (kπx/L) is handled similarly.

Euler-Fourier Formulas

Assuming f (x) defined on [−L, L] can be represented as a Fourier series we write

f (x) ∼ \frac{a_{0}}{2} + ∑_{n=1}^{∞}(a_{n} cos \frac{nπx}{L} + b_{n} sin \frac{nπx}{L}) ,

where

a_{0} = \frac{1}{L} ∫_{−L}^{L}f (x) dx
a_{n} = \frac{1}{L} ∫_{−L}^{L}f (x) cos \frac{nπx}{L} dx
b_{n} = \frac{1}{L} ∫_{−L}^{L}f (x) sin \frac{nπx}{L} dx

for n = 1, 2, . . ..