9&:4

한경대학교 공업수학 응용수학[ ](2) 2015 봄학기 안 상 욱[[email protected]]

In applied mathematics(2), we will refer to Fourier Anlysis, Partial Differential Equations, V ector Calculus, and Complex Analysis.

(A) Fourier analysis is composed of the following three parts:

(1) Fourier Series (2) Fourier Integral (3) Fourier Transfor m

A sequence is t he list of numbers looking like !_"# !_$# ⋯ # !_&# ⋯ where !_&

is called a general term of t he sequence, for example !_" is called the first term of the sequence, !_$ the second term, and so on. We often represent the sequence !_"# !_$# ⋯ # !_&# ⋯ as '!_&( )

You know that a sequence '!_&( creates the (infinite) Series

*

& + "

∞

!_&.

& +"

*

∞

!_&+ !_" - !_$ - ⋯ is defined by

*

& + "

∞

!_&+

lim

&→∞

2_& where 2_&+

*

3 + "

&

!₃.

Given any infinitely many differentiable f unction 4567, then Taylor series of 4567 centered at ! is defined by

4567 +

*

& +8

∞

9&:

4^5&75!7

56 ; !7^&

where 4⁵⁸⁷5!7 + 45!7# 4^5&7567 is & ; <= derivative of 4567,

&: + & ∙5& ; "7 ∙⋯ ∙$ ∙", and 8 ≠")

The Taylor series of 4567 cent ered at 8 is called Maclaurin series of 4567 and thus Maclaurin series of 4567 is defined by

4567 +_{& + 8}

*

^{∞ 945&7}_&:^5876&

(1)

A

^4′567 B567 C6 + 4567B567 ;

A

^{4567B′567 C6}

(2)

A

_!^D4′567B567 C6+ E4567 B567 F_!^D ;

A

_!^{D4567B′567 C6}

A function 4567 is called a periodic function if 456 - G7 + 4567 for every 6 in the domain of 4 and some positive number G and the least number of such G values is called a period of a function 4567)

We say that t wo functions 4567 and B567 are orthogonal on a closed interval E!# D F if

A

_!^D4567 B567 C6+ 8)

Let a function 4567 be a periodic function of a period $H.

Then

(1) " and IJ29H

&K

6 are orthogonal on E;H# HF .

(2) " and 23&9H

&K

6 are ort hogonal on E;H# HF.

(3) 23&9H

LK6 and IJ29H

&K

6 are orthogonal on E;H# HF.

(4) IJ29H

LK6 and IJ29H

&K

6 are orthogonal on E;H# HF 34 L≠&.

(5) 23&9H

LK6 and 23&9H

&K

6 are orthogonal on E;H# HF 34 L≠&.

Recall that (1) IJ25M ± O7 + IJ2MIJ2O ∓ 23&M 23&O (2) 23&5M ± O7 + 23&M IJ2O ± IJ2M 23&O

If a series !₈ -

*

& +"

∞

!_&IJ29H

&K

6 -

*

& + "

∞

D_&23&9H

&K

6 converges to a function 4567,

then (1) !₈+ 9$H

"

A

_;H^{H4567 C6}

(2) !_&+ 9H

"

A

_;H^{H4567IJ2 9}H

&K6 C6 if & ≥ " (called Euler’s formula)

(3) D_&+ 9H

"

A

_;H^H4567 23& 9H

&K

C6 if & ≥ "

and t hus

4567 + !₈ -

*

& +"

∞

!_&IJ29H

&K

6 -

*

& + "

∞

D_&23&9H

&K

6 if 4 is continuous at 6

and on the other hand if 4 is not continuous at 6, then a series

!₈ -

*

& + "

∞

!_&IJ29H

&K

6 -

*

& + "

∞

D_&23& 9H

&K

6 + 9$

456 - 87 - 456 ; 87

where 456 - 87 +

lim

<→6^-

45<7 and 456 ; 87 +

lim

<→6^;

45<7.

Now we are going to define a Fourier series of a periodic function 4567 on E; H# HF of a period R +$H by

4567 + !₈ -& + "

*

∞

!_&IJ2 9H

&K

6 - &+ "

*

∞

D_&23&9H

&K

6 if 4 is continuous at 6 and

9$

456 - 87 - 456 ; 87

+ !₈ -

*

& + "

∞

!_&IJ2 9H

&K

6 -

*

& + "

∞

D_&23& 9H

&K 6

if 4 is not continuous at 6

where (1) !₈ + 9$H

"

A

_;H^H4567C6

(2) !_&+ 9H

"

A

_{; H}^{H4567 IJ29&K}_H 6 C6 if & ≥" (called Euler’s formula)

Example 1: Find the Fourier series of a periodic f unction

4567 +

S

; T^T 34 ; K ' 6 '8^{34 8 ' 6 'K} # R + $K# T ≠8 Solution: since $H+ R + $K# H + K.

(1) !₈+ 9$K

"

A

; K K

4567 C6+ 9$K

"

5 ^A

;K

8; T C6 -

A

8 K

T C6

7

^{+ 8}

(2) & ≥ "# !_&+ 9K

"

A

_;K^{K4567 IJ29&K}_K ^{C6 + 9}_K^"

5 ^A

;K 8

; T IJ2 &6 C6 -

A

₈^KT IJ2 &6C6

7

+ 9K

"

5

^{; TUVW9}&

23&&6 X YZ

; K 8

- TU VW

9&

23& &6 X YZ

8

K

7

^{+ 9}K

"

58 - 87 +8

(3) & ≥"# !_&+ 9K

"

A

_{; K}^K4567 23&9K

&K

C6 + 9K

"

5 ^A

;K 8

; T 23& &6 C6 -

A

8 K

T 23& &6C6

7

+ 9K

"

5

^{; TUVW; 9}&

IJ2&6 X Y Z

;K 8

- TU V

W_{; 9}

&

IJ2 &6 X Y Z

8 K

7

+ 9&

$T

5

⁹" ; 5; "7_& ^&

7

^+[\

]

^

^

8 34 & 32 @_@&

9&K

`T 34 & 32 JCC

Hence 4567 + !₈ -

*

& + "

∞

!_&IJ29K

&K

6 -

*

& + "

∞

D_&23&9K

&K 6 +

*

& + "

∞

D_&23& &6

+

*

&+ "

∞

9K

$T

5

⁹&

" ; 5; "7^&

7

23& &6 + 9K

`T

5

23&6 - 9a

"

23&a6 - 9b

"

23&b6 - ⋯

7

Note that since 4567 is continuous at 9$

K, we have

T +4

5

⁹$ K

7

^{+ 9}K

`T

5

^{" ; 9}a

"

- 9b

"

; - ⋯

7

^{and so}

9`

K + " ; 9a

"

- 9b

"

; - ⋯ +

*

& + "

∞

5; "7& ;"

9$& ; "

"

+

*

& +8

∞

5; "7^&9$& - "

"

Note that for every nonnegative integer &, 23&&K + 8# and IJ2&K + 5; "7^&)

Note that if a periodic function 4567 on E; H# HF of a period R +$H has constants c and T satisfying

d 4567d ≤c@^T6 for every 6 in t he domain of 4

then the Fourier series of 4567 always exists.

Q1. (a) Find the Fourier seies of a periodic function

4567 +

S

^8# 34 ; K ' 6 ' 8

6^$ 34 8 ≤ 6 ' K # R + $K

(b) Use (a) t o find the sums of

*

& + "

∞

9&^$

"

and

*

& + "

∞

5; "7&; "

9&^$

"

.

Note t hat (1) The Fourier coefficients of a sum 4_" - 4_$ are the sums of the corresponding Fourier coefficients of 4_" and 4_$ respectively.

(2) The Fourier coefficient s of I4 are I times the corresponding Fourier coefficients of 4)

Q2. Find the Fourier series of the function

4567 + 6 - K 34 ; K '6 ' K !&C 456 - K7 +4567)

Q3. Find the Fourier series of the function

4567 + 23&K6 34 ; " ' 6 ' "# !&C R +$

Q4. Find the Fourier series of the function 4567 + IJ2K6 34 ; 9$

"

' 6 ' 9$

"

!&C R + "

Even and Odd f unctions

(1) We say that 4 is an even function if 45; 67 + 4567 if every 6 in the domain of 4) Note that t he graph of 4 is symmetric about f;axis and thus

we have

A

_{; !}^{!4567C6 + $}

A

8

!

4567 C6 if 4 is even.

(2) We say that 4 is an odd function if 45; 67 + ; 4567 if every 6 in the domain of 4) Note that the graph of 4 is symmetric about origin and thus we have

A

_{; !}^!4567 C6 +8 if 4 is odd.

Note that (1) The Fourier series of an even f unction 4 of period $H is a

4567 + !₈ -

*

& + "

∞

!_&IJ29H

&K

6 ( called a Fourier cosine series of 4)

wit h coefficients !8+ 9H

"

A

8 H

4567 C6# !&+ 9H

$

A

8 H

4567IJ2 9H

&K

6C6 34 & ≥")

(2) The Fourier series of an odd function 4 of period $H is a

4567 +

*

& + "

∞

D_&23& 9H

&K

6 ( called a Fourier sine series of 4)

with coefficients D_&+ 9H

$

A

8 H

4567 23&9H

&K

6C6 34 & ≥")

Half-Range Expansions

Let a function 4 is defined on only interval 58# H7)

(1) Then 4 can be extended to an even function 4_" of period $H where

4_"567 + 4567 34 8 ' 6 'H)

and the Fourier series of 4_" is a Fourier cosine series

4_"567 + !₈ - & + "

*

∞

!_&IJ29H

&K 6

wit h coefficients !₈+ 9H

"

A

₈^{H4567 C6# !&+ 9}_H^$

A

₈^{H4567IJ2 9&K}_H 6C6 34 & ≥")

Remember t hat if 8 ' 6 'H, then

4567 + 4_"567 + 4_"567 +!₈ - & + "

*

∞

!_&IJ2 9H

&K 6 (called an even periodic extension of 4)

wit h coefficients !₈+ 9H

"

A

8 H

4567 C6# !_&+ 9H

$

A

8 H

4567IJ2 9H

&K

6C6 34 & ≥")

(2) Then 4 can be extended to an odd f unction 4_$ of period $H where

4_$567 + 4567 34 8 ' 6 'H)

and the Fourier series of 4_$ is a Fourier sine series

4_$567 +

*

& + "

∞

D_&23&9H

&K 6

with coefficients D_&+ 9H

$

A

₈^H456723& 9H

&K 6C6.

Remember t hat if 8 ' 6 'H, then

4567 + 4_$567 +

*

& +"

∞

D_&23&9H

&K 6

( called an odd periodic extension of 4)

wit h coefficients D_&+ 9H

$

A

₈^H4567 23&9H

&K 6C6.

Q5. Find the even and odd periodic extentions of the function

4567 + [

\ ]

^ ^

9H

$T 34 8 ' 6≤ 9$

H

9H

$T5H ; 67 34 9$

H' 6 'H

where T ( 8)

Q6. Find t he even and odd periodic extentions of the f unction

4567 + [

\ ]

^ ^

6 34 8 ' 6 ≤ 9$

K

9$

K 34 9$

K'6 ' K

῿἟῿἟῿἟῿἟῿἟῿

Q7. Find t he even and odd periodic extentions of the function

4567 + 6^$ 34 8 ' 6'H

Q8. Find t he even and odd periodic extentions of the function

4567 + 6 34 8 ' 6'H

Q9. Find t he even and odd periodic extentions of the f unction

4567 + K ; 6 34 8 '6 ' K

Q10. Find t he even and odd periodic extentions of the f unction

4567 + 23&6 34 8 ' 6 ' K