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.
To consider this idea in more detail, we need to introduce some definitions and common terms.
Basic Definitions
A function
- Suppose that the function
with period is absolutely integrable on so that the following so-called Dirichlet integral is finite: - Suppose also that the function
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
At a discontinuity
The Fourier series of the function
where the Fourier coefficients
Sometimes alternative forms of the Fourier series are used. Replacing
we can write:
Fourier Series of Even and Odd Functions
The Fourier series expansion of an even function
where the Fourier coefficients are given by the formulas
Accordingly, the Fourier series expansion of an odd
where the coefficients
Below we consider expansions of
Solved Problems
Example 1.
Let the function
Calculate the coefficients
Solution.
To define
For all
Therefore, all the terms on the right of the summation sign are zero, so we obtain
In order to find the coefficients
The first term on the right side is zero. Then, using the well-known Product-to-Sum Identities, we have
if
In case when
Thus,
Similarly, multiplying the Fourier series by
Rewriting the formulas for
Example 2.
Find the Fourier series for the square
Solution.
First we calculate the constant
Find now the Fourier coefficients for
Since
Thus, the Fourier series for the square wave is
We can easily find the first few terms of the series. By setting, for example,
The graph of the function and the Fourier series expansion for