Calculus

Fourier Series

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Definition of Fourier Series and Typical Examples

Baron Jean Baptiste Joseph Fourier (1768 − 1830) introduced the idea that any periodic function can be represented by a series of sines and cosines which are harmonically related.

Baron Jean Baptiste Joseph Fourier (1768−1830)
Fig.1 Baron Jean Baptiste Joseph Fourier (1768−1830)

To consider this idea in more detail, we need to introduce some definitions and common terms.

Basic Definitions

A function f(x) is said to have period P if f(x+P)=f(x) for all x. Let the function f(x) has period 2π. In this case, it is enough to consider behavior of the function on the interval [π,π].

  1. Suppose that the function f(x) with period 2π is absolutely integrable on [π,π] so that the following so-called Dirichlet integral is finite:
    ππ|f(x)|dx<;
  2. Suppose also that the function f(x) is a single valued, piecewise continuous (must have a finite number of jump discontinuities), and piecewise monotonic (must have a finite number of maxima and minima).

If the conditions 1 and 2 are satisfied, the Fourier series for the function f(x) exists and converges to the given function (see also the Convergence of Fourier Series page about convergence conditions.)

At a discontinuity x0, the Fourier Series converges to

limε012[f(x0ε)f(x0+ε)].

The Fourier series of the function f(x) is given by

f(x)=a02+n=1{ancosnx+bnsinnx},

where the Fourier coefficients a0, an, and bn are defined by the integrals

a0=1πππf(x)dx,an=1πππf(x)cosnxdx,bn=1πππf(x)sinnxdx.

Sometimes alternative forms of the Fourier series are used. Replacing an and bn by the new variables dn and φn or dn and θn, where

dn=an2+bn2,tanφn=anbn,tanθn=bnan,

we can write:

f(x)=a02+n=1dnsin(nx+φn)orf(x)=a02+n=1dncos(nx+θn).

Fourier Series of Even and Odd Functions

The Fourier series expansion of an even function with the period of does not involve the terms with sines and has the form:

where the Fourier coefficients are given by the formulas

Accordingly, the Fourier series expansion of an odd -periodic function consists of sine terms only and has the form:

where the coefficients are

Below we consider expansions of -periodic functions into their Fourier series assuming that these expansions exist and are convergent.

Solved Problems

Example 1.

Let the function be -periodic and suppose that it is presented by the Fourier series:

Calculate the coefficients and

Solution.

To define we integrate the Fourier series on the interval

For all ,

Therefore, all the terms on the right of the summation sign are zero, so we obtain

In order to find the coefficients we multiply both sides of the Fourier series by and integrate term by term:

The first term on the right side is zero. Then, using the well-known Product-to-Sum Identities, we have

if

In case when , we can write:

Thus,

Similarly, multiplying the Fourier series by and integrating term by term, we obtain the expression for

Rewriting the formulas for we can write the final expressions for the Fourier coefficients:

Example 2.

Find the Fourier series for the square -periodic wave defined on the interval

Solution.

First we calculate the constant

Find now the Fourier coefficients for

Since we can write:

Thus, the Fourier series for the square wave is

We can easily find the first few terms of the series. By setting, for example, we get

The graph of the function and the Fourier series expansion for is shown below in Figure

Fourier series for the square  2pi-periodic wave
Figure 2, n = 10

See more problems on Page 2.

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