Lecture5. Fourier Series
In 1807 the French mathematician Joseph Fourier (1768-1830) submitted a paper to the Academy
of Sciences in Paris. In it he presented a mathematical treatment of problems involving heat conduction.
Although the paper was rejected for lack of rigor, it contained ideas that were so rich and widely
applicable that they would occupy mathematicians in research to the present day.
One surprising implication of Fourier's work was that many functions can be expanded infinite series or
integrals involving sines and cosines. This revolutionary idea sparked a heated debate among leading
mathematicians of the day and led to important advances in mathematics (Cantor's work on cardinals and
ordinals, orders of infinity, measure theory, real and complex analysis, differential equations), science and
engineering (data compression, signal analysis) , and applications undreamed of in Fourier's day (CAT
scans, PET scans, nuclear magnetic resonance).
A Fourier series is a representation of a function as a series of constants times sine and/o r cosine
functions of different frequencies. In order to see why such a series might be interesting, we will look at a
problem of the type that led Fourier to consider them.
1. Periodic Functions
A function f(x) is sa,d to be periodic if its function values repeat at regular intervals of the independent
variable. The regular interval between repetitions is the period of the oscillations.
Any periodic function f(x) is sometimes expressed as a series of a number of different sine components.
The component with the largest period is the first harmonic, or fundamental of f(x).
And in general formula, y = An sin(nx) is the nth harmonic, with amplitude An and period 2π/n.
The trigonometric functions sine and cosine are the simplest periodic functions. If we can express an
arbitrary periodic function in terms of these, many problems would be simplified.
2. The Fourier Series of a Function
𝐿
Let f(x) be defined for -L < x < L. For the time being, we assume only that −𝐿 𝑓(𝑥) 𝑑𝑥 exists. We want
to explore the possibility of choosing numbers a0, a l, . . . , bl, b2 , . . . such that
for -L < x < L. What does this tell us about how to choose the coefficients?
There is clever device used to answer this question, which was known to Fourier and others of his time.
The following identities will help us:
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The Fourier coefficients
Let n and m be nonnegative integers. Then
In the terminology of linear algebra, the trigonometric functions form an orthogonal coordinate basis. We
can easily prove these formulas if we remember the following trigonometric identities:
2 cosAcosB = cos(A - B) + cos(A + B)
2 sinAsinB = cos(A - B) - cos(A + B)
2 cosAsinB = sin(A + B) - sin(A - B)
Now, to find ao, integrate the series f(x) term by term:
All of the integrals on the right are zero and this equation reduces to
Therefore
Next, we will determine ak for any positive integer k. Multiply equation of series f(x) by cos(kπx/L) and
integrate each term of the resulting series to get
all of the integrals on the right are zero except for cos(nπx/L) cos(kπx/L)dx , which occurs when n = k, and
in this case this integral equals L. The right side of this equation therefore collapses to just one term, and
the equation becomes
whereupon
To determine bk, return to equation of f(x). This time multiply the equation by sin(kπx/L ) and integrate
each term to get
2
Similarly, all terms on the right are zero except for sin(nπx/L)sin(kπx/L)dx when n = k, and this equation
reduces to
Therefore
Example: Let f(x) = x for -π< x < π. Find the Fourier series expansion of f(x) at [-π, π].
The coefficients are
In this example the constant term and cosine coefficients are all zero, and the Fourier series contains only
sine terms.
Example:
Let f(x) = x2 for -L< x < L. Find the Fourier series expansion of f(x) at [-L, L] having period 2L.
The coefficients are
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Even and Odd Functions
Sometimes we can save some work in computing Fourier coefficients by observing special properties of
f(x). If f(x) is an even function on [-L, L], it has a property of f(-x) = f(x) for −𝐿 ≤ 𝑥 ≤ 𝐿. Similarly,
f(x) is an odd function on [-L, L], it has a property of f(-x) = -f(x) for −𝐿 ≤ 𝑥 ≤ 𝐿.
For example, x2, x4, cos(nπx/L) and e -IxI are even functions on any interval [-L, L]. The functions x, x3, x5
and sin(nπx/L) are odd functions on any interval [-L, L].
Graphs of even functions x2 and cos(5πx/3). They are symmetric about the y-axis
Graphics of odd functions y = x, y = x3, and y = sin(5πx/2) . They are symmetric through the origin.
Even and odd functions behave as follows under multiplication:
even •even = even ,
odd • odd = even,
even • odd = odd.
For example x2cos(nπx/L) is an even function (product of two even functions); x2sin(nπx/L) is odd
(product of an even function with an odd function); and x3sin(nπx/L) is even (product of two odd
functions). Now recall from calculus that
and
Example:
Find the Fourier series of f(x) = x4 on [-1, 1].
Since f is an even function, x4sin(nπx) is odd and we know immediately that all the sine coefficients bn,
are zero. For the other coefficients, compute:
and
4
The Fourier series of x4 on [-l, 1] is
3. The Fourier Cosine and Sine Series
If f(x) is defined on [-L, L], we may be able to write a series containing just cosines or just sines in
attempting to represent f(x) on the half-interval [0, L].
The Fourier Cosine Series of a Function
Let f(x) be integrable on [0, L]. We want to expand f(x) in a series of cosine functions. An even function
fe(x) is defined as
where fe(x) is an even function, fe(-x) = f(x), and agrees with f on [0,L] with fe (x) = f(x) for 0 < x < L.
We call fe the even extension of f to [-L, L]. Its Fourier series on [-L, L] is
in which
since fe(x) = f(x) for 0 < x < L.
We call the series above the Fourier cosine series of f(x) on [0, L]. The coefficients are the Fourier cosine
coefficients of f(x) on [0, L].
Example:
Let f(x) = e2X for 0≤ x ≤1. Find the Fourier cosine series of f(x).
The coefficients are
The cosine expansion of f is
The Fourier Sine Series of a Function
By duplicating the strategy just used for writing a cosine series, except now extending f(x) to an odd
function fo over [-L, L], we can write a Fourier sine series for f(x) on [0, L]. In particular, if f(x) is defined
on [0, L], let
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Then fo is an odd function, and fo (x) = f(x) for 0 ≤ x ≤ L. This is the odd extension of f to [-L, L].
Similiarly the Fourier series of fo is
With coefficient
Example:
Let f(x) = e2X for 0 ≤ 𝑥 ≤ 1. Find the Fourier cosine series of f(x).
The coefficients are
The sine series is
4. Complex Fourier Series and the Frequency Spectrum
It is often convenient to work in terms of complex numbers, even when the quantities of interest are real.
For example, electrical engineers often use equations having complex quantities to compute currents,
realizing that at the end the current is the real part of a certain complex expression.
We will cast Fourier series in this setting. Later, complex Fourier series and their coefficients will provide
a natural starting point for the development of discrete Fourier transforms.
Consider the Fourier series of f(x)
Using Euler's formula eix = cos x+i sin x, we can express the sine and cosine functions as:
Therefore
If we define c0 = a0 and
We will obtain
where
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For a function of period 2L we have
Example:
Find the complex Fourier series for the function 𝑓(𝑥) = 𝑥 if −𝜋 < 𝑥 < 𝜋 and 𝑓(𝑥 + 2𝜋) = 𝑓(𝑥).
We have to evaluate the integral
For n = 0 this integral is zero, so we have c0 = 0. For 𝑛 ≠ 0
Therefore
Note that we can obtain the real Fourier series from the complex one. If we
add nth and -nth terms we get
This is the real Fourier series.
Example:
Find the complex Fourier series of 𝑓 𝑥 = 𝑘.
If n = 0 we have
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Exercises
1. Find the Fourier series of the periodic function f(x) defined on the given interval.
Function
Answer
2. For the following functions defined on 0 < x < L, find the half-range cosine and half-range sine
expansions.
3. Find the complex Fourier series of the following functions:
References:
1. Advanced Engineering Mathematics, International Student Edition, Peter V. O'Neil.
2. Advanced Engineering Mathematics, K.A. Stroud, 4th edition.
3. Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson and S. J. Bence, 3th
edition.
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