This document discusses Fourier series and their applications. It contains the following key points: 1. Fourier introduced Fourier series to solve heat equations through metal plates, expressing functions as infinite sums of sines and cosines. 2. Sine and cosine functions are orthogonal and periodic, allowing any piecewise continuous periodic function to be represented by a Fourier series. 3. The Euler-Fourier formulas relate the Fourier coefficients to the function, allowing the coefficients to be determined. 4. Even functions only have cosine terms, odd only sine, and the Fourier series converges to the average at discontinuities for piecewise continuous functions.

1. Jose Stewart
Advised by Dr. Netra Khanal
The University of Tampa
April 2013
Fourier Series

2. Partial Differential Equations
 Differential equations arise in many areas of
science and technology, where continuously
variable quantities and their rates of change are
related.
 ODE-Equation for unknown function which relates
values of functions and its derivates of various
orders.
 PDE-Differential Equation involving multivariable
functions and their partial derivates.

3. Fourier Series
 Fourier introduced series for the purpose of
solving the heat equation through a metal plate
 Method involves expressing a function as an
infinite sum of sines and/or cosines
f(x)=
a0
2
+ (am cos
mpx
Lm=1
¥
å +bm sin
mpx
L
)

4. Periodicity of Sine and Cosine
 A function is periodic with period T>0 if the
domain of the function contains (x+T) when
containing x and if
 Also, if T is a period of a function, then any
integral multiple of T is also a period. The
smallest value of T for which the above equation
holds is called the fundamental period of the
given function.
f(x+T)= f(x)"x

5. Orthogonality of Sine and cosine
 The standard inner product (u,v) of two real
valued functions with (α ≤ x ≤ β) is given by
 The functions u and v are said to be orthogonal if
(u,v) = u(x)v(x)dx
a
b
ò
u(x)v(x)dx = 0
a
b
ò

6.  A set of functions is said to be mutually orthogonal if each distinct
pair of functions in the set is orthogonal.
 The functions sin(mx/L) and cos(mx/L), m=1,2,3… form a mutually
orthogonal set of functions on the interval –L<x<L. They satisfy the
following orthogonality relations.
cos
mpx
L-L
L
ò cos
npx
L
dx =
0, m ¹ n
L, m = n
ì
í
î
ü
ý
þ
cos
mpx
L-L
L
ò sin
npx
L
dx = 0, "m,n
sin
mpx
L-L
L
ò sin
npx
L
dx =
0, m ¹ n
L, m = n
ì
í
î
ü
ý
þ
These results can be obtained through direct integration. Consider equation 3.
(1
)
(2)
(3)

7. sin
mpx
L-L
L
ò sin
npx
L
dx =
1
2
cos
(m - n)px
L
-cos
(m + n)px
L
é
ëê
ù
ûú
-L
L
ò dx, m ¹ n{ }
Recall that
If m=n, then
Now, we have
sin
mpx
L-L
L
ò sin
npx
L
dx = (sin
mpx
L
)2
-L
L
ò dx =
1
2
1-cos
2mpx
L
é
ëê
ù
ûúdx
-L
L
ò
=
-L
L
1
2
x-
sin(2mpx /L)
2mp /L
ì
í
î
ü
ý
þ
= L.
2sinq1 sinq2 = cos(q1 -q2 )-cos(q1 +q2 )

8. Euler-Fourier Formulas
 Suppose that a series of the form
 The coefficients am and bm can be related to f(x) as a
consequence of the orthogonality conditions expressed in
equations (1), (2), and (3).
a0
2
+ (am
m=1
¥
å cos
mpx
L
+ bm sin
mpx
L
)
converges, call its sum f(x). Then
f (x) =
a0
2
+ (am cos
mpx
L
+
m=1
¥
å bm sin
mpx
L
)(4)

9. We will first multiply equation (4) by cos(nπx/L), where n is
a fixed positive integer and integrate with respect to x from
–L to L.
f (x)cos
npx
L-L
L
ò dx =
a0
2
cos
npx
L-L
L
ò dx + am cos
mpx
L-L
L
ò
m=1
¥
å cos
npx
L
dx + bm sin
mpx
L-L
L
ò
m=1
¥
å cos
npx
L
dx
Recall that n is fixed while m ranges over the positive integers. It follows from the
orthogonality equations (1) and (2) that the only non-zero term from equation 5 is
where m=n in the first summation. Thus,
(5)
f (x)cos
npx
L
ò dx = Lan, n =1,2,3.....
To find a0, we will integrate equation 4 from –L to L, yielding
f (x)dx =
a0
2-L
L
ò dx + am cos
mpx
L
dx
-L
L
ò + bm sin
mpx
L
dx = La0,
-L
L
ò
m=1
¥
å
m=1
¥
å
-L
L
ò
since each integral involving a trigonometric function is zero.

10. Consequently, we have
an =
1
L
f (x)cos
npx
L
dx, n = 0,1,2....
-L
L
ò
Similarly, we can express bn by multiplying equation 4 by sin(nπx/L)
and integrating term-wise from -L to L and using the orthogonality
relations (equations (2) and (3)). Then
bn =
1
L
f (x)sin
npx
L
dx, n =1,2,...
-L
L
ò
(6)
(7)
Equations (6) and (7) are known as Euler-Fourier for the coefficients in
a Fourier series.

11.  Example:
 Assuming there is a Fourier Series converging to the function f defined by
 We will determine the coefficients for the Fourier series.
 To determine a0 we use the Euler-Fourier formula where m=0
 For m>0, we have
f (x) =
-x, -2 £ x £ 0
x 0 £ x £ 2
ì
í
î
ü
ý
þ
f (x + 4) = f (x)
T = 4 Þ L = 2 Þ f (x) =
a0
2
+ am cos
mpx
2
+ bm sin
mpx
2
é
ëê
ù
ûú
m=0
¥
å
a0 =
1
2
(-x)dx +
1
2
xdx =1+1= 2
0
2
ò
-2
0
ò
am =
1
2
(-x)cos
mpx
2
dx +
1
2-2
0
ò xcos
mpx
2
dx
0
2
ò

12. 1
2
(-x)cos
mpx
2-2
0
ò dx,let u = -x dv = cos(
mpx
2
)
Þ du = -dx, v = sin(
mpx
2
)(
2
mp
)
Þ
1
2
(-x)cos
mpx
2-2
0
ò dx =
1
2
-(
2
mp
)xsin(
mpx
2
)-cos(
mpx
2
)(
2
mp
)2é
ëê
ù
ûú
-2
0
=
1
2
-(
2
mp
)2
+ cos(mp)(
2
mp
)2é
ëê
ù
ûú

13. 1
2
xcos(
mpx
20
2
ò )dx,let u = x dv = cos(
mpx
2
)
Þ du = dx, v = sin(
mpx
2
)(
2
mp
)
Þ
1
2
xcos(
mpx
2
)dx =
0
2
ò
1
2
(
2
mp
)xsin(
mpx
2
) + cos(
mpx
2
)(
2
mp
)2é
ëê
ù
ûú
0
2
=
1
2
cos(mp)(
2
mp
)2
- (
2
mp
)2é
ëê
ù
ûú

14. So then
am =
1
2
-(
2
mp
)2
+ cos(mp)(
2
mp
)2
+ cos(mp)(
2
mp
)2
-(
2
mp
)2é
ëê
ù
ûú
=
1
2
-
4
(mp)2
+ 2cos(mp)(
4
(mp)2
) -
4
(mp)2
é
ë
ê
ù
û
ú
=
4
(mp)2
(cos(mp) -1), m =1,2,3,....
=
-8
(mp)2
, m = 2n -1(n Î N)
0 m = 2n
ì
í
ï
îï
ü
ý
ï
þï

15.  Similarly,
bm =
1
2
-xsin(
mpx
2
)dx +
1
2-2
0
ò xsin(
mpx
20
2
ò )dx
1
2
-xsin(
mpx
2-2
0
ò )dx,let u = -x dv = sin(
mpx
2
)dx
Þ du = -dx, v = -cos(
mpx
2
)(
2
mp
)
Þ
1
2
xcos(
mpx
2
)(
2
mp
)
é
ëê
ù
ûú
-2
0
+ (
2
mp
) cos(
mpx
2-2
0
ò )dx
=
1
2
xcos(
mpx
2
)(
2
mp
) + sin(
mpx
2
)(
2
mp
)2é
ëê
ù
ûú
-2
0
= cos(mp)(
2
mp
)

16. 1
2
xsin(
mpx
20
2
ò )dx,let u = x dv = sin(
mpx
2
)dx
Þ du = dx, v = -cos(
mpx
2
)(
2
mp
)
Þ -xcos(
mpx
2
)(
2
mp
)
é
ëê
ù
ûú
0
2
+
1
2
-cos(
mpx
20
2
ò )(
2
mp
)dx
=
1
2
-xcos(
mpx
2
)(
2
mp
) + sin(
mpx
2
)(
2
mp
)2é
ëê
ù
ûú
0
2
= -cos(mp)(
2
mp
)
Þ bm = cos(mp)(
2
mp
) -cos(mp)(
2
mp
) = 0, "m

17. f (x) =
a0
2
+ (am cos
mpx
L
+
m=1
¥
å bm sin
mpx
L
)
So then
=1-
8
p2
cos
px
2
+
1
32
cos
3px
2
+
1
52
cos
5px
2
+ ....
æ
è
ç
ö
ø
÷
=1-
8
p2
cos(mpx /2)
m2
m=1,3,5...
¥
å
f (x) =1-
8
p2
cos((2n -1)px /2)
(2n -1)2
n=1
¥
å

18. Even and Odd functions
 It is useful to recognize two classes of functions for
which the aforementioned Euler-Fourier formulas can
be simplified. These are even and odd functions and
they are delineated geometrically by the property of
symmetry with respect to the y-axis and the origin.
 A function f is an even function if its domain contains
the point –x whenever it contains the point x, and if
 Examples of even functions are 2, x4, cos(nx), x2n.
f (-x) = f (x)

19. f is said to be an odd function if its domain contains –x
whenever it contains x and if
f (-x) = -f (x)
The functions x, x3, x2n+1 and sin(nx) are examples of odd
functions. Most functions are neither even or odd.
Elementary properties of even and odd functions include the
following:
1. The sum and product of two even functions are even.
2. The sum of two odd functions is odd. The product of two
odd functions is even.
3. The sum of an odd and even function is neither even
nor odd. The product of two such functions is odd.

20.  Perhaps of greater importance are the following integral
properties. If f is an even function, then
(8)
If f is an odd function, then:
It is important to also discuss cosine and sine series as
they lend themselves towards determining a Fourier
series.
f (x)dx = 2 f (x)dx
0
L
ò
-L
L
ò
f (x)dx = 0
-L
L
ò(9)

21. Cosine Series
 Suppose that f and f’ are continuous on –L ≤ x ≤ L and f is
a periodic even function with period 2L. From properties
1 and 3 we have f(x)cos(nπx/L) is even and f(x)sin(nπx/L)
is odd. Consequently, from equations (8) and (9) the
Fourier coefficients of f are given as
Thus, f has the Fourier series
So we have established that the Fourier series of any even function
consists only of the even trigonometric functions cos(nπx/L) and the
constant term.
Such a series is called a Cosine series.
an =
2
L
f (x)cos
npx
L
dx , n = 0,1,2,...
0
L
ò bn = 0 , n =1,2,....
f (x) =
a0
2
+ an cos
npx
Ln=1
¥
å

22. Sine Series
 Supposing that f and f’ are continuous on
-L ≤ x ≤ L and that f is an odd function of period 2L.
Properties 2 and 3 dictate that f(x)cos(nπx/L) is odd and
f(x)sin(nπx/L) is even; in which case the Fourier coefficients
of f are
Yielding the Fourier series
Thus the Fourier series for any odd function consists of the
odd trigonometric functions sin(nπx/L). This type of series
is called a Fourier Sine Series.
an = 0 , n = 0,1,2,....
bn =
2
L
f (x)sin
npx
L
dx , n =1,2,3,....ò
f (x) = bn sin
np x
Ln=1
¥
å

23. Fourier Convergence Theorem
 We have established that if the Fourier series
converges and defines a function f, then the coefficients am
and bm are related to f(x) by the Euler-Fourier formulas:
a0
2
+ [am cos
mpx
Lm=1
¥
å + bm sin
mpx
L
]
am =
1
L
f (x)cos
mpx
L
dx, m = 0,1,2....
-L
L
ò
bm =
1
L
f (x)sin
mpx
L
dx, m =1,2,...
-L
L
ò

24.  A function is said to be piecewise continuous on an interval
a ≤ x ≤ b if the interval can be partitioned by a finite number of points a= x0 <
x1 <…<xn = b so that
1. f is continuous on each open subinterval xi-1 < x <xi
2. f approaches a finite limit as the endpoints of each subinterval are
approached from within the subinterval.
Theorem: Suppose that f and f’ are piecewise continuous
on the interval –L ≤ x < L. Furthermore, suppose that f is
defined outside the given interval so that it is periodic
with period 2L. Then f has a Fourier series
The Fourier series converges to f(x) where f is continuous
and to [f(x+) + f(x-)]/2 at all points where f is
discontinuous.
f (x) =
a0
2
+ [am cos
mpx
Lm=1
¥
å + bm sin
mpx
L
]

25.  Consider
f (x) =
0 -L < x < 0
L 0 < x < L
ì
í
î
ü
ý
þ
, f (x +2L) = f (x)
f (x) =
L
2
+
2L
p
sin((2n-1)px / L)
2n-1n=1
¥
å
a0 = L
am = 0 , m ¹ 0
bm =
0, m even
2L
mp
m odd
ì
í
ï
î
ï
ü
ý
ï
þ
ï

27. Fourier Transform
 The Fourier transform transforms a function from
one of time to a function of frequency, which is
reversible.
 Most often, the original function involved is real
valued and the transform yields a complex valued
function.
ˆf (x) = f (t)e-2pitx
-¥
¥
ò dt
f (t) = ˆf (x)e2pixt
dx
-¥
¥
ò

28.  Linearity: For any complex numbers a and b, if
 Translation:
h(t)= af (t)+bg(t)
then ˆh(x) = aˆf (x)+bˆg(x)
h(x) = f (x - x0 ), then ˆh(x) = e-2pix0x ˆf (x)
x0 Î Â

29.  Derivatives:
dˆf
dt
= (2pix) ˆf (x)
d2 ˆf
dt2
= (2pix)2 ˆf (x)
dn ˆf
dtn
= (2pix)n ˆf (x)

30. Some useful Fourier Transforms
Convolution:
( f *g)(t) = f (t -t)g(t)dt
-¥
¥
ò = f (t)g(t -t)dt
-¥
¥
ò

31. Consider the following Differential Equation:
¢¢y (t)- y(t) = -g(t)
Þ (2pix)2
ˆy(x)- ˆy(x)= -ˆg(x)
Þ ˆy(x) 4p2
i2
x2
-1éë ùû= - ˆg(x)

32. Þ ˆy(x) =
ˆg(x)
1+ 4p2
x2
Þ y(t) = (
ˆg(x)
1+ 4p2
x2
)Ú
Þ y(t) =
e
-t
2
*g(t)
y(t) =
1
2
e-(t-t )
g(t)dt
-¥
¥
ò

33. Questions?