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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

(2)

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

(3)

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

(4)

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)

(5)

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

(6)

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

(7)

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

  

 

  

  

  

(8)

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

(9)

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

(10)

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)

n

a

n

n

  

(11)

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)

n

a

n

n

  

(12)

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)

n

a

n

n

  

(13)

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)

n

a

n

n

  

2 1

1 ( 1) 1

{ cos sin

) 4

( }

n

n

nx nx

n n

f x

      

(14)

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)

n

a

n

n

  

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( , )

(15)

Vector Space & Function Space

(16)

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

(17)

n-Dimensional Euclidean Space

T

( , ) a ba ba b 1 1   a b n n

R

n

with the inner product

is called the n-dimensional Euclidean space and is denoted by E

n

or 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

(18)

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

(19)

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

) ( )

(

(20)

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

)

~ ( )

~ (

(21)

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

)

~ ( )

~ (

~

(22)

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

(23)

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

(24)

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, vu vu vu v

Properties

) , ( )

( u, vv u

scalar a

: , ) , ( )

( k u, vk v u k

0 if

, 0 )

( and 0

if , 0 )

( u, uuu, uu

) (

) (

)

( uv, wu, wv, w

(25)

Orthogonal Functions

Inner Product of Functions

The inner product of two functions and on an interval is the number Definition 12.1

f

1

b

a

f x f x dx f

f , ) ( ) ( )

(

1 2 1 2

f

2

[ b a , ]

Orthogonal Functions

Two functions and are said to be orthogonal on an interval if Definition 12.2

f

1

1 2 1 2

( , )

b

( ) ( ) 0 f f  

a

f x f x dx

f

2

[ b a , ]

(26)

Orthogonal Functions

Orthogonal Functions

Two functions and are said to be orthogonal on an interval if Definition 12.2

f

1

0 )

( ) ( )

,

( f

1

f

2

 

ab

f

1

x f

2

x dx

f

2

[ b a , ]

2

1

x

ff

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

(27)

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

b

a m n

n

m

( ), ( ))   ( ( ) ( ))0 ,

(    

0

( x ),

1

( x ),

2

( x ),

] , [ b a

Norm of a function

b

a 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 ,...

(28)

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

n

x dx n

 

n m

dx x x

n

m

 

( ) ( ) 0 ,

 

0 ,

0 )]

sin(

1 [sin 1 sin

cos )

( ) ( )

,

(

0 0

n n

n n n nx

nxdx dx

x x

n

n

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

n

m n

m

 

 

 

, 0 ) ]

sin(

) [ sin(

2 1

] ) cos(

) [cos(

2 1

cos cos

) ( ) ( )

, (

 

In the second case,

(29)

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 )

2

dx 2

0

( ) x 2

  

2 2

( ) cos

1 [1 cos 2 ] 2

n

x nxdx

nx

  

Thus for

 

 0 , ( x )

n

n

b

a n n

n n

n

( x ) (  ,  ) (  ( x )  ( x )) dx

( ) cos , 0

n

x nx n

  

(30)

Orthogonal Functions

Orthogonal Series Expansion

(31)

Orthogonal Functions

Orthogonal Series Expansion

  

 ( ) ( ) ( )

)

( x c

0 0

x c

1 1

x c x

f  

n

n

Q) 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

(32)

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 0

x c

1 1

x c x

f  

n

n

Q) 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

(33)

Orthogonal Functions

 By utilizing the inner product

By orthogonality, zero except when mn

  

   

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

     

b

f ( x )

n

( x ) dx c

n b

2n

( x ) dx

 

b n

b a n

dx x x

f

c ( ) ( )

Orthogonal Series Expansion

  

 ( ) ( ) ( )

)

( x c

0 0

x c

1 1

x c x

f  

n

n

Q) 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

(34)

Orthogonal Functions

 

b

n 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 mn

  

   

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

     

(35)

Fourier Series

 

b

a 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

 

b

a 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 0

x c

1 1

x c x

f  

n

n

(36)

Fourier 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 )

(37)

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

m n

n n p n p

xdx x dx x x

p p n p

  

 

      

 

 

)

,

(  p p

참조

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