Ряды Фурье_4курс

1. Ряды Фурье

2.

a
g (t ) 0
2
ak cos(k 1t ) bk sin(k 1t )
k 1
1 2 / T
ak
2
bk
T
2
T
T /2
g (t ) cos(k t )dt
1
T / 2
T /2
g (t ) sin(k t )dt
1
T / 2
a0 1
2 T
T /2
g (t )dt
T / 2

3.

2
ak
T
1
t'
1
ak
T /2
2
1
g (t ) cos(k
t )dt
T
T / 2
T'
g (t ) cos(k
T '
T /2
g (t ) cos(k
T / 2
2
2
t ) d (t )
T
T
2
2
t )d ( t)
T
T
2
t, T '
T
1
ak
g (t ) cos(kt ' )d (t ')
g (t ) cos(kt)dt,
k 0,1,2,...
a
g (t ) 0
2
1
bk
g (t ) sin(kt )dt
ak cos(kt ) bk sin(kt )
k 1
k 0,1,2,...

4.

2
ak
bk
2
g (t ) cos(kt )dt,
k 0,1,2,...
0
g (t ) sin (kt )dt,
0
k 0,1,2,...

5.

a
g (t ) 0
2
Ak cos(k 1t k )
k 1
Ak ak2 bk2 , cos k ak / Ak ,
a
g (t ) 0
2
sin k bk / Ak ,
ak cos kt bk sin kt
k 1
a
g (t ) 0 2
2
~
Ak cos(kt k )
k 1
~ 1
Ak
ak2 bk2 ,
2
cos k ak / 2 Ak ,
sin k bk / 2 Ak
~
Ak Ak / 2

6.

e jx cos x j sin x
cos x
a
g (t ) 0
2
g (t )
k 1
(3.15)
1 jx
(e e jx )
2
(
3.16)
Ak
exp( jk 1t j k ) exp( jk 1t j k )
2
C k e jk 1t
(3.18)
k
g (t )
2 jkt
C k exp
T
k
1
Ak e j k
2
Ak 2 C k , k arg(C k )
(3.19)
Ck
(3.20)
(3.17)

7.

g (t )
C k e jkt
k
~ j k
C k Ak e
(3.21)
~
Ak Ak / 2
g (t )
k
g (t )
1
Ak e j ( k 1t k )
2
~ j ( kt k )
Ak e
(3.23)
(3.24)
k
g (t )
~
Ak e jk (t k )
k
(3.25)
k k k
(3.22)

8.

(
ak
b
j k
2
2
3.26)
ak 2 Re(C k ), bk 2 Im(Ck )
Ck
k arctg (bk / ak )
(
3.27)
*
C k C k
1
Ck
T
Ck
1
T
(3.29)
T /2
g (t )e jk 1t dt
T / 2
(
3.30)
T /2
2 jkt
g (t ) exp
dt
T
T / 2
(3.31)