Fourier Series

1
Fourier Series
2
We will see that many important problems
involving partial differential equations can be
solved, provided a given function can be
expressed as an infinite sum of sines and
cosines.
In this section and in the following two
sections, we explain in detail how this can be
done.
These trigonometric series are called Fourier
series, and are somewhat analogous to Taylor
series, in that both types of series provide a
means of expressing complicated functions in
terms of certain familiar elementary functions.
3
Periodic Function
The function f(t) defined for all t is said to be
periodic provided that there exists a positive
number p such that
for all t. The number p is then called a period
of the function f.
Note The period of a periodic function is not
unique, for example, if p is period of f(t), then
so are the numbers 2p, 3p, and so on
4
Example
The period of the functions
(where n is a positive integer) is 2?/n because
and
Moreover, 2? itself is a period of the functions
g(t) and h(t).
5
Fourier Series Representation of Functions
We begin that every function f(x) with period 2?
can be represented by an infinite trigonometric
series of the form
6
On the set of points where this series converges,
it defines a function f whose value at each point
x is the sum of the series for that value of x.
In this case the series is said to be the Fourier
series of f.
Our immediate goals are to determine what
functions can be represented as a sum of Fourier
series, and to find some means of computing the
coefficients in the series corresponding to a
given function.
7
Orthogonality of Sine and Cosine
The functions sin(m? x/L) and cos(m? x/L), m
1, 2, , form a mutually orthogonal set of
functions on -L ? x ? L, with
8
Finding Coefficients in Fourier Expansion
Suppose the series converges, and call its sum
f(x). Let f(x) be defined on the closed interval
-? x ?
The coefficients an, n 1, 2, , can be found as
follows.
Multiplying (1) by cosmx, where m is any fixed
ve integer, and integrate from -? to ?
9
Integrating term by term, we see that the right
side becomes
10
We know that
11
These facts enable us to
and hence
12
We get the corresponding formula for bn by
essentially the same procedure we multiply (1)
through by sinmx and integral term-by-term
Now the coefficient a0
13
and hence
Thus the coefficients are given by the equations
which known as the Euler-Fourier formulas.
14
A Fourier series is thus a special kind of
trigonometric series one whose coefficients are
obtained by applying formulas (2) and (3) to some
given function f(x). In order to form this
series, it is necessary to assume that f(x) is
continuous, but only the integrals (2) and (3)
exist and for this it suffices to assume that
f(x) is integrable on the interval -? x ?.
15
Problem 1. Find the Fourier Series for the
function defined by
Solution We first find the coefficient a0, an, bn
We know that
16
Now
and hence
and
17
(No Transcript)
18
(No Transcript)
19
Problem 5(d). Find the Fourier series for the
function
Solution We first find the coefficient a0, an, bn
We know that
20
For n 1
21
Similarly, we can show
and
22
The problem of convergence.
Recall that the function f is said to be
piecewise continuous on the interval a, b
provided that there is finite partition of a, b
with endpoints
Such that

• f is continuous on each open interval ti-1lttltti.
• At each endpoint ti of such a subinterval the
  limit of f(t), as t approaches ti from within the
  subinterval, exists and is finite.

23
It follows that a piecewise continuous function
is continuous except possibly at isolated points,
and that at each such point of discountinuity,
the one-sides limits
Thus a piecewise continuous function has only
(isolated ) finite jump discontinuities like the
one shown in fig.
24
Dirichlet Theorem
Let f be a piecewise smooth function in -?, ?.
That is f be a piecewise continuous first
derivative on -?, ?. Then the Fourier series
expansion for f converges pointwise everywhere in
-?, ?, and has the value
25
at each point x0 in the interior of the interval
and
at ??.
(i.e. the average of the right and left-hand
limits of f at x0 and is equal to f(x0), whenever
x0 is a point of continuity of f.)
26
1
3?
?
2?
-?
-2?
-3?
-1
27
Problem 3 page
Find the Fourier series for the periodic function
defined by
Find what numerical sums are implied by the
convergence behavriour at point of discontinuity
x0 and x?
28
Solution
We first find the coefficient a0, an, bn
Now
29
so
30
and hence Fourier Series is
By Dirichlets Theorem this equation isvalid at
all points of continuity since f(x) is
understood to be periodic extension of the
initially given point
31
At the point of discountity x ?, the series
When x ?, we get
At the point of discountity x 0, the series
When x 0, we get
32
?
3?
?
2?
-?
-2?
-3?
-?