Naval Architecture & Ocean Enginee ring
Engineering Mathematics 2
Prof. Kyu-Yeul Lee
Department of Naval Architecture and Ocean Engineering, Seoul National University of College of Engineering
[2008][09-2]
October, 2008
Naval Architecture & Ocean Enginee ring
Fourier Series(1)
: Fourier, Sine and Cosine Series
Orthogonal Functions and Fourier Series Fourier Series
Fourier Cosine and Sine Series
Complex Fourier Series
Preview
0
1
0
( ) cos sin
2 ,
1 ( )
1 ( ) cos 1 ( )sin
n n
n
p p p
n p
p
n p
a n n
f x a x b x
p p
where
a f x dx
p
a f x n xdx
p p
b f x n xdx
p p
Fourier Series
Preview
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)Preview
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)0, 0
) , 0
(
f x
x x
x
Ex.) Expand
in a Fourier series
Preview
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)0, 0
) , 0
(
f x
x x
x
Ex.) Expand
in a Fourier series
x
y
2
Preview
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)0, 0
) , 0
(
f x
x x
x
Ex.) Expand
in a Fourier series
x
y
2
0
0 0
2 0
1 1
[ 0 ( )
1 [ ]
2 2
( )
a f dx dx x
x x
x dx
Preview
0
1
0
( ) cos sin
2 ,
1 ( )
1 ( ) cos 1 ( )sin
n n
n
p p p
n p
p
n p
a n n
f x a x b x
p p
where
a f x dx
p
a f x n xdx
p p
b f x n xdx
p p
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)0, 0
) , 0
(
f x
x x
x
Ex.) Expand
in a Fourier series
x
y
2
0
2
a
Preview
0
1
0
( ) cos sin
2 ,
1 ( )
1 ( ) cos 1 ( )sin
n n
n
p p p
n p
p
n p
a n n
f x a x b x
p p
where
a f x dx
p
a f x n xdx
p p
b f x n xdx
p p
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)0, 0
) , 0
(
f x
x x
x
Ex.) Expand
in a Fourier series
x
y
2
0
2
a
0
0
0 0
1 cos
1 [ 0 ( )cos ]
1 sin 1
[( ) ( 1)sin ]
1 cos cos 1 1 ( 1
( )
)
n
n
a nxdx
dx x nxdx
x nx nxdx
n n
nx n
f x
Integrating by part
Preview
0
1
0
( ) cos sin
2 ,
1 ( )
1 ( ) cos 1 ( )sin
n n
n
p p p
n p
p
n p
a n n
f x a x b x
p p
where
a f x dx
p
a f x n xdx
p p
b f x n xdx
p p
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)0, 0
) , 0
(
f x
x x
x
Ex.) Expand
in a Fourier series
x
y
2
0
2
a
2
1 ( 1)
na
nn
Preview
0
1
0
( ) cos sin
2 ,
1 ( )
1 ( ) cos 1 ( )sin
n n
n
p p p
n p
p
n p
a n n
f x a x b x
p p
where
a f x dx
p
a f x n xdx
p p
b f x n xdx
p p
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)0, 0
) , 0
(
f x
x x
x
Ex.) Expand
in a Fourier series
x
y
2
0
2
a
0
0
0 0
1 sin
1 [ 0 ( )sin ]
1 cos 1
[ ( ) cos ]
1 1 1
si (
n
n
)
b nxdx
dx x nxdx
x nx nxdx
n n
nx
f x
Integrating by part
2
1 ( 1)
na
nn
Preview
0
1
0
( ) cos sin
2 ,
1 ( )
1 ( ) cos 1 ( )sin
n n
n
p p p
n p
p
n p
a n n
f x a x b x
p p
where
a f x dx
p
a f x n xdx
p p
b f x n xdx
p p
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)0, 0
) , 0
(
f x
x x
x
Ex.) Expand
in a Fourier series
x
y
2
0
2
a 1
b
n n
2
1 ( 1)
na
nn
Preview
0
1
0
( ) cos sin
2 ,
1 ( )
1 ( ) cos 1 ( )sin
n n
n
p p p
n p
p
n p
a n n
f x a x b x
p p
where
a f x dx
p
a f x n xdx
p p
b f x n xdx
p p
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)0, 0
) , 0
(
f x
x x
x
Ex.) Expand
in a Fourier series
x
y
2
0
2
a 1
b
n n
2
1 ( 1)
na
nn
2 1
1 ( 1) 1
{ cos sin
) 4
( }
n
n
nx nx
n n
f x
Preview
0
1
0
( ) cos sin
2 ,
1 ( )
1 ( ) cos 1 ( )sin
n n
n
p p p
n p
p
n p
a n n
f x a x b x
p p
where
a f x dx
p
a f x n xdx
p p
b f x n xdx
p p
Fourier Series
0
1
0
cos sin
2 , 1
1 cos
1 si
( )
( ) ( ) ( ) n
n n
n
p p p
n p
p
n p
f x
f x f x f x
a n n
a x b x
p p
where
a dx
p
a n xdx
p p
b n xdx
p p
( ) : Given
f x
approximation for f(x)0, 0
) , 0
(
f x
x x
x
Ex.) Expand
in a Fourier series
x
y
2
0
2
a 1
b
n n
2
1 ( 1)
na
nn
2 1
1 ( 1) 1
{ cos sin
) 4
( }
n
n
nx nx
n n
f x
approximation for f(x)
( ) ( , )
S x on S x on( ) ( , ) S ( )x on( , )
Vector Space & Function Space
Inner Product Spaces
b a
b a b
a T ( , )
Inner Product (dot product)
n n
b b a
a
1 1
n
l
l l b a
1
n n b a b
a
1 1
The length or norm of a vector in V is defined by
( , )
a a a 0
n-Dimensional Euclidean Space
T
( , ) a b a b a b 1 1 a b n n
R
nwith the inner product
is called the n-dimensional Euclidean space and is denoted by E
nor again simply by R
n.
Euclidean norm
T 2 2
( , ) a 1 a n
a a a a a
) tors column vec
: ,
where
( a b
Inner Product for Functions. Function Space
The set of all real-valued continuous function f (x), g(x), ∙∙∙ on a given interval α ≤ x ≤ β is a real vector space under the usual addition of functions and multiplication by scalars (real numbers).
On this “function space” we can define an inner product by the integral
( , ) f g f x g x dx ( ) ( )
n n n
l
l
l
b a b a b
a
1 1
1 T
b ( a , b ) a b
a
Inner Product for Functions. Function Space
The set of all real-valued continuous function f (x), g(x), ∙∙∙ on a given interval α ≤ x ≤ β is a real vector space under the usual addition of functions and multiplication by scalars (real numbers).
On this “function space” we can define an inner product by the integral
( , ) f g f x g x dx ( ) ( )
n n n
l
l
l
b a b a b
a
1 1
1 T
b ( a , b ) a b a
N
i
i
i x dx g x dx
f
1
) ( )
(
Inner Product for Functions. Function Space
The set of all real-valued continuous function f (x), g(x), ∙∙∙ on a given interval α ≤ x ≤ β is a real vector space under the usual addition of functions and multiplication by scalars (real numbers).
On this “function space” we can define an inner product by the integral
( , ) f g f x g x dx ( ) ( )
n n n
l
l
l
b a b a b
a
1 1
1 T
b ( a , b ) a b a
N
i
i
i x dx g x dx
f
1
) ( )
(
N
i
i
i x g x
f
1
)
~ ( )
~ (
Inner Product for Functions. Function Space
The set of all real-valued continuous function f (x), g(x), ∙∙∙ on a given interval α ≤ x ≤ β is a real vector space under the usual addition of functions and multiplication by scalars (real numbers).
On this “function space” we can define an inner product by the integral
( , ) f g f x g x dx ( ) ( )
n n n
l
l
l
b a b a b
a
1 1
1 T
b ( a , b ) a b a
N
i
i
i x dx g x dx
f
1
) ( )
(
N
i
i
i x g x
f
1
)
~ ( )
~ (
~
Inner Product for Functions. Function Space
norm
( , ) f g f x g x dx ( ) ( )
( , ) ( ) 2 .
f f f f x dx
Our examples give a first impression of the great generality of the abstract concepts of vector space and function space
)
~ ( )
~ (
x g x
f
n n n
l
l
l
b a b a b
a
1 1
1 T
b ( a , b ) a b a
( , )
a a a
Inner Product for Functions
Orthogonal Functions
Introduction
In certain area of advanced mathematics, a functions is considered to be a generalization of a vector
How the two vector concepts of inner product and
orthogonality of vectors can be extended to functions
Orthogonal Functions
Inner Product of vectors
k v j v i v k
u j
u i
u
1
2
3,
1
2
3 v
u
3 3 2
2 1
)
1( u, v u v u v u v
Properties
) , ( )
( u, v v u
scalar a
: , ) , ( )
( k u, v k v u k
0 if
, 0 )
( and 0
if , 0 )
( u, u u u, u u
) (
) (
)
( u v, w u, w v, w
Orthogonal Functions
Inner Product of Functions
The inner product of two functions and on an interval is the number Definition 12.1
f
1
ba
f x f x dx f
f , ) ( ) ( )
(
1 2 1 2f
2[ b a , ]
Orthogonal Functions
Two functions and are said to be orthogonal on an interval if Definition 12.2
f
11 2 1 2
( , )
b( ) ( ) 0 f f
af x f x dx
f
2[ b a , ]
Orthogonal Functions
Orthogonal Functions
Two functions and are said to be orthogonal on an interval if Definition 12.2
f
10 )
( ) ( )
,
( f
1f
2
abf
1x f
2x dx
f
2[ b a , ]
2
1
x
f f
2 x
3[ 1 , 1 ]
Ex)
6 0 ) 1
, (
1
1 1 6
1
3 2 2
1
x x dx x
f
f orthogonal
Orthogonal Functions
Orthogonal Set
A set of real-valued functions is said to be orthogonal on an interval if
Definition 12.3
n m
dx x
x x
x
ba m n
n
m
( ), ( )) ( ( ) ( )) 0 ,
(
0( x ),
1( x ),
2( x ),
] , [ b a
Norm of a function
ba n n
n n
n
( x ) ( , ) ( ( x ) ( x )) dx
Any orthogonal set of nonzero functions can be normalized by dividing by its norm
n( x ) , n 0 , 1 , 2 ,...
Orthogonal Functions and Fourier Series
Example 1
Orthogonal Set of Functions
Show that the set is orthogonal on the interval
} , 2 cos , cos , 1
{ x x
] , [
nx x
x ) 1 ,
n( ) cos
0
(
0 ,
0 )
( )
0
(
x
nx dx n
n m
dx x x
nm
( ) ( ) 0 ,
0 ,
0 )]
sin(
1 [sin 1 sin
cos )
( ) ( )
,
(
0 0
n n
n n n nx
nxdx dx
x x
nn
In the first case,
n n m
m
x n m n
m
x n m
dx x n m x
n m
nxdx mx
dx x x
nm n
m
, 0 ) ]
sin(
) [ sin(
2 1
] ) cos(
) [cos(
2 1
cos cos
) ( ) ( )
, (
In the second case,
Orthogonal Functions and Fourier Series
Example 2 Norms
Find the norms of each function in the orthogonal set given in Example 1
1 )
0
( x
0( x )
2dx 2
0
( ) x 2
2 2
( ) cos
1 [1 cos 2 ] 2
n
x nxdx
nx
Thus for
0 , ( x )
n
n
ba n n
n n
n
( x ) ( , ) ( ( x ) ( x )) dx
( ) cos , 0
n
x nx n
Orthogonal Functions
Orthogonal Series Expansion
Orthogonal Functions
Orthogonal Series Expansion
( ) ( ) ( )
)
( x c
0 0x c
1 1x c x
f
n
nQ) Is it possible to determine a set of coefficient ? c
n
n(x ) An infinite orthogonal set of function on interval a, b
) (x
f A function defined on interval a, b
Orthogonal Functions
Orthogonal Series Expansion
By utilizing the inner product
)) ( ),
( ( ))
( ),
( ( ))
( ),
( (
) ( ) ( )
( ) ( )
( ) (
) ( ) (
1 1 0
0
1 1
0 0
x x
c x
x c
x x
c
dx x x
c dx
x x
c dx x x
c
dx x x
f
m n
n m
m
m b
a n n
m b
m a b
a
m b
a
( ) ( ) ( )
)
( x c
0 0x c
1 1x c x
f
n
nQ) Is it possible to determine a set of coefficient ? c
n
n(x ) An infinite orthogonal set of function on interval a, b
) (x
f A function defined on interval a, b
Orthogonal Functions
By utilizing the inner product
By orthogonality, zero except when m n
dx x x
c dx
x x
c dx x x
c
dx x x
f
m b
a n n
m b
m a b
a
m b
a
) ( ) ( )
( ) ( )
( ) (
) ( ) (
1 1
0
0
bf ( x )
n( x ) dx c
n b
2n( x ) dx
b nb a n
dx x x
f
c ( ) ( )
Orthogonal Series Expansion
( ) ( ) ( )
)
( x c
0 0x c
1 1x c x
f
n
nQ) Is it possible to determine a set of coefficient ? c
n
n(x ) An infinite orthogonal set of function on interval a, b
) (x
f A function defined on interval a, b
Orthogonal Functions
bn n b
a
n
w x x dx
dx x x
w x c f
) ( )
(
) ( ) ( ) (
2
Orthogonal Set/ Weight Function
A set of real-valued functions is said to be orthogonal with respect to a weight function on an interval if Definition 12.4
n m
dx x x
x
b
w
a m n
( ) ( ) ( ) 0 ,
0( x ),
1( x ),
2( x ),
] , [ b a )
(x w
By utilizing the inner product
By orthogonality, zero except when m n
dx x x
w x c
dx x x
w x c
dx x x
w x c
dx x x
w x f
m b
a n n
m b
m a b
a
m b
a
) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) (
) ( ) ( ) (
1 1
0
0
Fourier Series
ba n
n b
a
n
x dx
dx x x
c f where
) (
) ( ) , (
2
Orthogonal series expansion of (generalized Fourier Series)
0
) ( )
(
n
n
n
x
c x
f
ba n
n b
a
n
w x x dx
dx x x
w x f c
or
) ( )
(
) ( ) ( ) (
,
2 )
(x f
Assumption : an orthogonal set is complete.
If is a set of real-valued functions that is orthogonal on an interval and if is a function defined on the same interval. then
can be formally expanded in an orthogonal series
0( x ),
1( x ),
2( x ),...
) (x ] f
, [ b a )
(x f
( ) ( ) ( )
)
( x c
0 0x c
1 1x c x
f
n
nFourier Series
Fourier Series
The Fourier series of a function defined on the interval Is given by
Definition 12.5
p n p
p n p
p p
n
n n
p xdx x n
p f b
p xdx x n
p f a
dx x p f
a
where
p x b n
p x a n
x a f
sin ) 1 (
cos )
1 (
) 1 (
,
sin 2 cos
) (
0
1 0
f ( p , p )
Fourier Series
p n p
p n p
p p
n
n n
p xdx x n
p f b
p xdx x n
p f a
dx x p f
a where
p x b n
p x a n
x a f
sin ) 1 (
cos ) 1 (
) 1 (
,
sin 2 cos
) (
0
1 0
2 3 2 3
1,cos x ,cos x ,cos x ,...,sin x ,sin x ,sin x ,...
p p p p p p
The set of trigonometric functions
is orthogonal on the interval ( p , p )
Show the given set of function is orthogonal (step.1)
sin n , 1, 2,3...;[0, ]
x n p
p
0 0
0 0
For :
1 ( ) ( ) ( ) ( )
sin sin cos cos sin sin 0
2 2( ) 2( )
p p
p p
m n
n m n m n m p n m p n m
x xdx x x dx x x
p p p p n m p n m p
2
0 0
0 0
For :
1 1 2 1 2
sin cos sin
2 2 2 4 2
p p
p p