, . , , . , . ,
, ,
.
; - , ; ω - , , 2 π ( ); φ - .
ω, 2ω,, k ω, , ,
, .
, , , , , , , , .
= .
bn, n =1,2,3,..., . ( ) .
, f (x), =2π, , ..
.
:
1. f (x) ?
2. ?
3. ?
.
4.4. 2π
.
( )
f (x) 2 π, - [- π; π ] , ..
,
:
,, ,, b , = 1,2,3,..., f (x) , , f (x).
, f (x), , , .
f (x) = 2π [- π; π ] , [ - π; π ] , f (x) , f (x) , f (x) S (x) f (x), f (x)
|
|
( ), ..
f (x), ( ), - [- π; π ].
,
.
, f (x) = 2 π , f (x), f (x). , .
, . . .
f (x) :
1) f (x) ;
2) f (x);
3)
4)
, .
, , f (x).
,
sin0 = 0, sin n π =0, cos0 = 1, cos n π = (-1) .
30
f (x + 2 π) = f (x).
.
f (x) f (x + 2 π)= f (x), , = 2 π. f (x) ( 3).
3
f (x) [- π; π ] , .. -.
:
,
.
, S (x) f (x) , .. , . = 0, = π, = 2 π,... ( ), .. x - t π, t Î Z.
.
,
.
, , 1/2( 4).
4
4.5. =2π
. f (x) [- π, π ] , .. Î D (y) Î D () f (- x) = f (x).
f (x) - , f (x) cosnx , f (x) sinnx . ,
|
|
:
,
,
bn = =0.
, -
a 0, an
, .
f (x) [- π. π ] , .. Î D (y) - Î D (y) f (- x) = - f (x), f (x) sin nx: , f (x) cos nx . , :
,
,
bn = .
, -
bn = .
31
f (x), ( 5).
.
, -2π, , . [- π,π ] , , .
5
, 0 = =0. b :
b :
, S (x) f (x), , .. ≠ tπ, t ÎZ,
.
, :
.
6 S (x) .
6
4.6. = 2 l
f (x) = 2 l [- l, l ] . .
. x = at, f (at) . , 2 π, .. . x = at = 2 π. , :
,
, , , t , , , dt = f (x)
,
, l = π, = 2 π.
f (x) = 2 1 , f (x)
,
0,
, .
f (x) - =21 , f (x)
bn .
32
= |sin x |.
( 7). , = |sin x | = π (, 2 l = π, l = π / 2), ( ). [0, ] , , (, ), , , ( ).
7
:
,
,
.
, , = | sinx | .
|
|
4.7. ,
f (x) [ a, b ] , [ a, b ] , f (x) . , .
f *(x) = 2 l ≥ b -, [- l, l ] [ a, b ]( 8).
8
f * () . f * (). [ a, b ] f (x) f * (), [ a, b ], , , , f (x) [ a, b ]
f (x) [0, l ] - , . [- l, 0], [- l, 0] f (- x) = f (x) f (- x)=- f (x). , . .
33
: .
.
[0 ,π ]. , , [0 ,π ] . , [- π,0], U * (ω t) = 2π ( 9),
9
U * (ω t) , , b =0. ωt =, a O, n:
,
, :
,
.
U * (ω t), . U (ω t) U * (ω t) ω t Î [ 0, π ],
.
. , . , .
10
, f (x) ( 10), [-1,1] =1 Î[-1,0) , (0,1) (1,2). , ( 1, 1) ( 2, 2),
.
, (0,1) (1,2):
.
, [-1,1]
, f (x) =2,.. f (x + 2 )= f (x).