08 seires fourier.pdf mm
DESCRIPTION
YiyhTRANSCRIPT
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pi
,
,
pi ,
,
, 38334
7 2014
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1 . 31.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 . 42.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 . 63.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 . . . . . . . . . . . . . . . . . . . . . . . 6
3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 . 84.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2.1.i , , pi . . . . . . . . . . 114.2.1.ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2.2 - . . . . . . . . . . . . . . . . . . . . . . . . . 184.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.1.i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.2 - . . . . . . . . . . . . . . . . . . . . . . . . . 364.3.2.i Duhamel. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 - , . 465.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.1.i Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.1.ii Neumann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.1.iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.1.iv . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2.2.i Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2.2.ii Neumann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.2.2.iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
ii
-
iii
5.3.1 Laplace. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3.1.i Dirichlet. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.4 - - pi. . . . . . 88
6 Fourier 906.1 Fourier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.1.1 , , Fourier. . . . . . . . . . . . . . 916.2 Fourier . . . . . . . . . . . . . . . . . . . . . . . 93
6.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.3 , . . . . . . . . . . . . . . . . . . . . . 95
6.2.3.i Fourier . . . . . . . . . . . . . . . . 956.2.3.ii Fourier . . . . . . . . . . . . . . . . . 96
6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.3.3 Fourier. . . . . . . . . . . . . . . . . . . 106
6.3.3.i Fourier. . . . . . . . . . . . . . . . . . . . . . . . . 1066.3.3.ii Fourier. . . . . . . . . . . . . . . . . . . . . . . . 108
6.4 Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.4.1 pi . . . . . . . . . . . . . . . . . . . . . . . . 1106.4.2 Fourier pi. . . . . . . . . . . . . . . . . . . 111
6.4.2.i Fourier pi. . . . . . . . . . . . . . . . 1116.4.2.ii Fourier pi. . . . . . . . . . . . 1126.4.2.iii Fourier pi. . . . . . . . . . . 1126.4.2.iv Fourier. . . . . . . . . . . . . . . 1126.4.2.v . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.4.2.vi Fourier. . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4.3 Fourier. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.4.4 Fourier . . . . . . . . . . . . . . . . . . . . . 1186.4.5 Fourier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.4.6 Gibbs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Fourier. . . . . . . . . . . . . 1216.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.5.2 Fourier. . . . . . . . . . . . . . . . . . . . . . . 127
7 - , pi . 1297.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.3 - -pi . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.4 pi . . . . . . . . . . . . . . . . . . . . . . . . 135
7.4.1 pi . . . . . . . . . . . . . . . 141
8 Sturm-Liouville. 1518.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.2 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.3 - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1518.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9 1529.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 152
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iv
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pi pi pi pi
:
: () () ()
:
:
:
:
-: -
:
..:
..:
.L.: Laplace
1
-
6
Fourier
6.1 Fourier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.1.1 , , Fourier. . . . . . . . . . . . . 91
6.2 Fourier . . . . . . . . . . . . . . . . . . . . . . 936.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.3 , . . . . . . . . . . . . . . . . . . . . 95
6.2.3.i Fourier . . . . . . . . . . . . . . . 956.2.3.ii Fourier . . . . . . . . . . . . . . . . 96
6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.3.3 Fourier. . . . . . . . . . . . . . . . . . 106
6.3.3.i Fourier. . . . . . . . . . . . . . . . . . . . . . . . 1066.3.3.ii Fourier. . . . . . . . . . . . . . . . . . . . . . . 108
6.4 Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.4.1 pi . . . . . . . . . . . . . . . . . . . . . . . 1106.4.2 Fourier pi. . . . . . . . . . . . . . . . . . 111
6.4.2.i Fourier pi. . . . . . . . . . . . . . . 1116.4.2.ii Fourier pi. . . . . . . . . . . 1126.4.2.iii Fourier pi. . . . . . . . . . 1126.4.2.iv Fourier. . . . . . . . . . . . . . 1126.4.2.v . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.4.2.vi Fourier. . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4.3 Fourier. . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.4.4 Fourier . . . . . . . . . . . . . . . . . . . . 1186.4.5 Fourier. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206.4.6 Gibbs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.5 Fourier. . . . . . . . . . . . 1216.5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.5.2 Fourier. . . . . . . . . . . . . . . . . . . . . . 127
90
-
6.1. Fourier. 91
6.1 Fourier.
. (5) pi - pi Dirichlet, Neumann, , pi pi Fourier pi, pi pipi (0, ), (,). pi pipi pi pi . pi pi pi
pi ; , (); pi pi pi Fourier pi-
; pi pipi pi pi pi .
pi pi Fourier pi ; , pi, pi Fourier pi.
pipi, cos sin ( pi ) pi pi pi pi pi pi Fourier. , pi pi pi pi Fourier pi pi - . pi pi [16], [10] [8].
6.1.1 , , Fourier.
d2
d2+ = 0
- , ,
sin(), / cos
()
Dirichlet, Neumann, . pi pi :
Fourier. Dirichlet pi -, (), 0 < < pi pi Fourier,
() =
=1
sin(), (6.1.1)
=
0
sin( )()
0
sin( )2
=2
0
sin(
)(), = 1, 2, . . . , (6.1.2)
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92 6. Fourier
Fourier. Neumann pi , (), 0 < < pi pi Fourier,
() = 0 +=1
cos(), , (6.1.3)
0 =1
0
() (6.1.4)
=2
0
cos(
)() =
0
cos( )()
0
cos( )2
, = 1, 2, . . . , (6.1.5)
Fourier. pi -, (), < < pi pi Fourier,
() = 0 +
=1
cos()
+
=1
sin()
(6.1.6)
,
0 =1
2
() (6.1.7)
=
() cos(
)
cos2(
)
=1
() cos(
) (6.1.8)
=
() sin(
)
sin2(
)
=1
() sin(
) (6.1.9)
Fourier pi - Fourier .
pi pi pi Fourier pi pi pipi pi Fourier pi . pi pi Fourier pi .
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6.2. Fourier . 93
6.2 Fourier .
Fourier pi pi pipi pi Fourier pi pipi , (0, ) (,).
6.2.1 .
pi .
6.2.1 ( ). () < < pi, pi > 0,
( + ) = (), (,). (6.2.1) pi ().
pi pi pi pi . , pi pi R. , pi, sin pi (,), pi (0, 2).
pi pi , :
) ( + ) = () ,
) pi pi pi pi . ,
) pi , +
() pi .
pi
pi 6.2.2 ( Fourier). pi Fourier (6.1.6) (pi pi pi ) pi pi pi 2, pi pi2 . pi Fourier, pi pi -pi.
, Fourier, ,
0 =1
2
()
pi pi pi () [,] pi () pi pi () 2.
cos(
) (,] -
sin(
) pi ,
pi Fourier pi . pi pi pi Fourier .
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94 6. Fourier
6.2.2
. pi 0 , (,) , pi
() = (), (,) (6.2.2) pi pi- pi 0.
, pi pi pi cos, (,)
. pi 0 , (,) pi, pi
() = (), (,) (6.2.3), pi pi pi pi 180 pi , pi . pi pi pi pipi pi 0 pi(0) = 0. . pipi pipipi pi pi pi pipi 0 pipi , < 0 pi . pipi pi pi , , pi .
, pi pi pi - pi sin, (,)
. pi. pi- () pi pi () pi :
() =1
2[() + ()] + 1
2[() ()] + (6.2.4)
= 12 [() + ()] = 12 [() ()]. pi () = ()() = ()
. = pi pi = pi =pi + = pi + pi =pi + pi pi .
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6.2. Fourier . 95
.
() , dd pi
() pi, dd
() , 0
() pi
() pi, 0
()
() pi,
() = 0
() ,
() = 20
()
, pi, pipi , dd pi pipi
dd (0) = 0.
pi pi pi Fourier.
6.2.3 , .
pi pi pi Fourier .
6.2.3.i Fourier .
pi pi pi:
pi Fourier pi pi Fourier
, pi pi (6.1.7-6.1.9) pi () = pi,
0 =1
2
() = 0,
=1
() cos(
) = 0
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96 6. Fourier
pi pi, cos(
)
pi . , () = pi, pi
Fourier( () pi) ==1
sin(), (,),
=1
() sin(
) =
2
0
() sin(
)
pi pipi pi () sin(
) .
pi sin pi , pi , pi . pipipi pi pi (,) (0, ) pi Fourier pi .
pi , (,) pi 1. pi 2 pi 2 ,
2. pi, pi sin( )
(,).
6.2.3.ii Fourier .
, pi, pi pi:
pi Fourier Fourier
, pi pi (6.1.7-6.1.9) pi () = ,
=1
() sin(
) = 0
pi pi, sin(
) pi
pi . , () = , pi
Fourier( () ) = 0 +=1
cos(), (,),
0 =1
2
() = 21
2
0
() =1
0
(),
=1
() cos(
) =
2
0
() cos(
)
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6.2. Fourier . 97
pi 0 pipi pi () () sin(
) . , pi, (pi ) cos
(
), pi cos
(
) .
pi, pi (,) (0, ) pi Fourier.
pi pipi pi (,) pi
1. pi 2 pi 2 ,
2. , pi, cos( ).
. pipi, pi () pi-, pi pi pi pi pi (6.1.7-6.1.9). pi . pi - , (); pi !. pi, pi
0 +
=1
cos(),
0 =1
2
()
=1
() cos(
)
pi pi
1
2
() = 1
0
()
1
() cos(
) = 2
0
() cos(
)
Fourier (), Fourier ().
pi pi pi pi Fourier (), Fourier ().
pi (,) pi () pi pi pi pi. pi- pi , pi pi
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98 6. Fourier
. pi pi - pi pi , ()
() = () + () =1
2[() + ()] + 1
2[() ()]
pi Fourier ()
() = 0 +
=1
cos()
+
=1
sin()
pi pi pi (6.1.7-6.1.9). , pi
0 =1
2
() =1
2
(() + ())
=1
2
() =1
0
=1
0
1
2[() + ()] (6.2.5)
pi
() = 0, pi 12
() =1
0
.
,
=1
() cos(
) =
1
cos(
)(() + ())
=1
cos(
)() =
2
0
cos(
)() =
=2
0
cos(
) 12
[() + ()] (6.2.6)
cos(
)() pi pi pi-
(,) cos ( ) () (,) pi (0, ). , pi
=1
() sin(
) =
1
sin(
) 12
[() ()] (6.2.7)
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6.2. Fourier . 99
pi 0 Fourier = 12 [() + ()], () = 12 [() ()] pipi pi . pi pi pipi , pi pipi pi .
pi: pi Fourier () Fourier , , Fourier pi , .
6.1: pi Fourier 2-pi
() = || . () pi (6.1). pi, pi , Fourier pi .
0 =1
0
() =1
0
=2
0
() cos () =2
0
cos ()
pi 0 || = . ,
0 =1
22
0
=
2
=2
sin
0
2
0
sin
=
2
cos
2
0
=2
(1) 12
, sin = 0 cos = (1). (1) 1 pi
(1) 1 ={2, pi
0,
Fourier
2 4
=pi
1
2cos =
2 4
=1
cos(2 1)(2 1)2 (6.2.8)
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100 6. Fourier
6.1: () = || ,
6.2: pi Fourier 2-pi
() = . () pi pi (6.2). pi, pi pi , Fourier pi .
=2
0
() sin () =2
0
sin ()
,
= 2
cos
0
+2
0
cos
= 2
[(1)] + 2
2sin
0
,
= 2(1)
= 2(1)+1
, sin = 0, cos = (1). Fourier
2=1
(1)+1
sin (6.2.9)
Bessel. pi Fourier pi pi: 6.2.3 ( Bessel). () pi pi 2 [,]
|0 |2 +1
2
1
(| |2 + | |2
) 1
2
|()|2 (6.2.10)
pi 0 , , , = 1, 2, . . . pi Fourier, pi pi pi (6.1.7 - 6.1.9)
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6.3. . 101
6.2: () = ,
pi. pi.
Fourier pipi Bessel pi pi pi Fourier, , , = 1, 2, . . . . pi (6.2.10) pi pi .
pi Fourier pi.
6.2.4 ( ): pi pi pi Fourier (,+) pi pi (,) (0, ). pipi pipi . pi:
pi Fourier (pi) (,+).
6.3 .
Fourier pi pi. pipi pi , pi Fourier pi pi ! pi, pi Fourier pi pi . pi pi pi pi .
Fourier pi pi. pi .
6.3.1
pi , , pi pi pi ( Fourier). pi pi pi pi pi pi, pi pi () pi , pi pipi pi pi.. C(1) .
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102 6. Fourier
pi pi =1
()
. pi . 6.3.1 ( ).
=1
() () (, )
()
=1
()
0 , (, ) (6.3.1)
() (, ). pi , pi . pi pi pi .
6.3.2 ( ).
=1
() () [, ]
max
()
=1
()
0 (6.3.2)
pi, pi pi [, ], . pi , pi .
pi Weierstrass -: pi
| | , =1
-
6.3. . 103
pi. ,
, 2 ,
;
2 ;
pi, pi pi pi . pi - pi . pipi, pi , pi pi pipi pi pi pi .
6.3.2 .
pi Fourier pi pi Fourier pi pi pi , pi pi pi pi pipi. pi pi . 6.3.4 ( ). () [, ] < < < . ()
1. [, ] pi pi pi 1 , 2 , . . . , .
2. pi , 1 , 2 , . . . , , pi pi :
() = lim0
( ) = (+) = lim0
( + ) =
> 0. pi pi pi pi .
[, ] (, ). R, R
pi [, ] R. pi pi
pi pi . 6.3.5 ( ). () [, ] < < < . () pi pi, () [, ]. ,
1. (, )2. pi (, ) pi pipi ( pi
) pi pi pi (pi ) .
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104 6. Fourier
[, ] (, ). R, R
pi [, ] R. pipi pi, pipi
() ( pi ) pipi pi ( pi ).
. pi . Fourier :
() = 0 +=1
[ cos
()
+ sin()]
(6.3.5)
6.3.6 ( Fourier). 2pi R
() pi pi (6.3.5)
lim
() =1
2[() + (+)]
. () (+) pi pi lim
() = ()
pi .
pi. pipi, (6.1) pi pi- .
pipi Fourier . 6.1 ( Dirichlet).pi
() pi
() =1
2
[1 + 2
=1
[cos(
)cos(
)+ sin
(
)sin(
)]]() (6.3.6)
pi (6.1.7), (6.1.8), (6.1.9) Fourier. pi pi (9.1.4) pi (-9)
cos(
)cos(
)+ sin
(
)sin(
)= cos
[
( )]
pi pi
() =1
2
( )() (6.3.7)
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6.3. . 105
pi
() = 1 + 2
=1
cos()
(6.3.8)
pi Dirichlet. pi pi pi 2. pi
1
2
() = 1 + 0 + 0 + . . . + 0 = 1 (6.3.9)
pi pi pi
() =sin[( + 12)
]
sin(12) (6.3.10)
pi = pi pi pi pi,
() =1
2
()( + ) (6.3.11)
,
() () = 12
() [( + ) ()] =1
2
() sin
[( +
1
2)
] (6.3.12)
() =( + ) ()
sin(12) (6.3.13)
pipi pi () pi (6.3.12) .
pipi pi pipi , pi .
pi pi pi pi pi, pi(6.1) pi pi Fourier . , pi (6.2) pi pi = pi, pi () = (+) = , 12 [() + (+)] = = 0. Fourier = . pipi Fourier .
pi 2-pi , pi pi Fourier , pi. pi pi pi pi
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106 6. Fourier
6.3.7 ( Fourier): pi 2-pi Fourier = .
pi. Fourier.
6.3.3 Fourier.
Fourier pi pi pi =1
(), ,
d
d
( =1
()
)=
=1
d()
d
pi pipi pi pi pi ( pi pi ) pi pipi pi pi . () pipi pi pi pi pi pi (). Fourier. , , Fourier pi pi.
6.3.3.i Fourier.
pi pi pi (pi, pi !) . , pi pi pi pi pi pi pi pi.
pi pi , , pi Fourier . 6.3.8 ( Fourier ). , 2-pi. Fourier . pi Fourier pi , ,
. ,
=
,
=
pi. pi pi pi.
=1
() cos(
) =
1
() cos(
)
1
()(
) sin(
)
= (
)
1
() sin(
) =
, () = () cos() = cos() = (1). pi pi pi.
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6.3. . 107
pi pi pi pi pi pi pi .
pi
6.3.9 ( Fourier)., , 2-pi. pi . ,
0 +=1
[ cos
()
+ sin()]
, pi pi pipi pi pi pi , pi
=1
[(
) cos
() (
) sin
()]
pi. pi pi Fourier (6.3.6) ( pi pi -). pi, pi pi pi . pipi pipi .
(6.3.6) pi pi pi pi Fourier.
pi 6.3.10 ( Fourier ). Fourier pi pi pi pi () .
pi pi pi Fourier pi (6.4.4).
. pi pi (6.3.9) , pi pi pi ., weierstrass - (6.3.3) Fourier pi pi
cos() | |
sin
() | |
(, , - ) ,
=0
| |
-
108 6. Fourier
2-pi, , Fourier pi .
pipi pi pi . pi ! pi pi pi Fourier pi pi pi- . pi pi pi, Fourier pi ||. pi pipipi pi pi pi pi . pi pi pi C(k1) pi (1) Fourier pi
2 0 > 1, C(k).
pi ( Fourier ):pi, pi, pi pi pi pi Fourier . , pi Fourier pi pipi .
Fourier pi pi pi () = pi Fourier pi 1, Fourier () = || pi , pi Fourier 2.
6.3.3.ii Fourier.
pi Fourier pi pi pi . pi, () = 1 pi, , () = pi . pi Fourier pi pi pi 0 . pi pi , pi
6.3.11 ( Fourier). 2-pi Fourier .
pi, () =0
(). , 0 = 0 () pi
Fourier pi pipi pi pi Fourier ().
() = 0 +
=1
[(
) sin
() (
) cos
()]
(6.3.16)
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6.4. Fourier . 109
pi 0 pi [,].
0 =1
2
() (6.3.17)
, 0 = 0 (6.3.16) pi pi ()0.pi. , . pi, 0 = 0, 2-pi ,
( + 2) () =+2
() =
() = 20 = 0
, pi (6.3.6) pi () Fourier . , pi (6.3.8) pi .
pi pi pi pi pi Fourier . pi pi pipi . pi pi pipi pi pipi . , . pi .
Fourier pi pi pi Fourier pi , , 2.
pi pi pi pi Fourier pi .
() .F. (), () .F. (), () pipi .F. 1
2[(+) + ()] ,
pi .. Fourier.
6.4 Fourier .
pi pi Fourier pi pi , pi pipi pipi pi pi (5). pi Fourier pi pi pi , pipi , pi . pipi : (0, ) (,] pipi pi (5). pi pi pi Fourier pipi :
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110 6. Fourier
: pi pi pi - R, Fourier pi . ,pi pi pipi pi pi .
6.4.1 pi .
pipi pi pi R. pi pipi ( pi pi pi ) . pi pi pi pi pi . pi pi pi pi, pi pi pi pi pi pi Fourier.
pi. pi pi pi-, , pi pi - pi pi pi , pi <
-
6.4. Fourier . 111
pi. () [0, ] pi pi [,] pi:
=
{(), 0 (), 0 (6.4.1)
pi (). pi pi . pi pi pi . , [0, ] pi .
pi. () (0, ] pi pi pi pi (,). pi pi ,
=
(), 0 < (), < 00, = 0.
(6.4.2)
pi pi (). pi pi pi pipi pipi = 0. pi pi pi pi pi pi , pi 180. pi, pi pi pi pi pi . , [0, ] pi pi .
pi pi pi.
pi: pi pi . (0, ], pi (,] pi 2.
pi.: pi pi . [0, ], pi [,] pi 2.
6.4.2 Fourier pi.
pi pipi pi pi, pi- pi, pi pi pi pi pi Fourier pi .
6.4.2.i Fourier pi.
(,] pi pi pi pi Fourier R pi [,] pi pi .
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112 6. Fourier
6.4.2.ii Fourier pi.
[0, ], pi [,], pi, pi pi 2-pi R. pi ( ) Fourier. pi pi ( pi pi) pi pi pi pi pi (6.2.3.ii) Fourier . pi . pi pi
pi ( Fourier pi): Fourier pi pi Fourier pi pi pi [0, ].
6.4.2.iii Fourier pi.
[0, ], pi pi [,], pi, pi pi pi 2-pi R. pi ( ) Fourier. pi pi pi ( pi pi) pi pi pi pi pi pi (6.2.3.i) Fourier pi .pi . pi pi
pi ( Fourier pi): Fourier pi pi Fourier pi pi pi [0, ].
6.4.2.iv Fourier.
pi pi pi () [0, ], pi, pi pi, pi , . :
6.4.1 ( Fourier). () [0, ].
0 +
=1
cos(), 0 =
1
0
(), =2
0
() cos()
Fourier .
=1
sin(), =
2
0
() sin()
Fourier .
pi pi [0, ] - Fourier .
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6.4. Fourier . 113
6.4.2.v .
pi pi Fourier , pipi pi , pi, , (6.3.6). pi pi pi pi pi pi , pi , pi, pi, pi, pi pi, .
pi: pi pi pi pi [,], pi pi pi , pi , pipi (jumps) = (2 + 1), , . , [,], pi = (2+ 1) () = (). pipi , pipi pi pi (2 + 1) ()(+) = () . , pi (6.2) [, ] () = () pi pi pi . , (,], pi pi R.
pi: [0, ], pi pi pi pi pi R. , , , pi pipi pipi . , [0, ] pi pi pi, pi 2 (2+1) (0+) = 0 () = 0 . pi pi pi pi : 2 (2 + 1) (0) = 0 () = 0 . pi .*
pi pipi , pi ((,] [0, ]) pipi (6.3.6) Fourier. , pi , pi , , pi pi pi pi .
6.4.2 ( Fourier ). (,] [0, ] , Fourier: ,, , 12 [(+)+()] (,) (0, ). () (,) (0, ), pi () . pi, (0+) = 0 () = 0 . 6.4.3: , pi (6.3.6) ((0, ) (,)) . 6.4.4 ( ): pi pipi Fourier
* pi pi pi pi pi- pi pi pi pi pi pi .
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114 6. Fourier
(, ] [0, ] , (,] [0, ] pi pi -. pi
=
(6.4.3)
pi () pi 2 () pi 2 . ,
() = () = (
) (6.4.4)
, pi pipi (, ) (0, ). ,pi = ( pi pipi ) (, + ) pi () = ( + )
6.4.2.vi Fourier.
pi Fourier, pi pipi .
Fourier pi. (6.4.2) - Fourier pi pi (), (,] pi :
1. (), (,].2. pi pi ().
3. pi , pi pi pi pipi .
pipi pi pi Fourier , pi Fourier pi pi .
Fourier. (6.4.2), Fourier (), [0, ] :
1. (), [0, ].2. pi () (,].3. pi pi pi ( pi 2).
4. pi , pi pi pi pipi .
, Fourier (), [0, ] :
1. (), [0, ].
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6.4. Fourier . 115
2. pi pi () (,].3. pi pi pi pi ( pi 2).
4. pi , pi pi pi pi- pipi .
6.3: pi Fourier () = , pi , [0, ].
pi. pi (6.4.2). - pi pi
=2
0
sin() =
2
0
sin() = 2
cos()
0
=2
[1 cos()] = 2
[1 (1)] =
{0, 4 , pi
(6.4.5)
,
=4
[sin()
+1
3sin
(3
)+
1
5sin
(5
)+ . . .
]=
4
pi.
1
sin()
(6.4.6)
pi pi. pi pi .
pi - [0, ] = 0 = pi , . , pi pi pi (6.3.2).
, pi pi pi 0 1 (0, ) pi pipi .
pipi pi pi pi pi . pi = 1, = = 2 ,
1 =4
pi.
1
sin(
2
)=
4
pi.
1
(1)(1)/2 = 4
=0
(1)2 + 1
(6.4.7)
pipi
4= 1 1
3+
1
5 1
7+
1
9+ . . . (6.4.8)
6.4: pi Fourier () = , pi , [0, ].
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116 6. Fourier
pi. pi (6.4.2). , , = 0 pi pi
=2
0
cos() =
2
0
cos() =
2
sin()
0
=2
[sin() sin(0)] = 0, = 0 (6.4.9)
(6.4.10)
,
0 =
0
= (6.4.11)
, pi pi = + 0 + 0 + . . .!
! pi pi pi pi : Fourier Fourier () pi , Fourier Fourier () ( pi [0, ].)
pi !! pi pi pi -
. , , pi pi pi pi (5) . , pi pi, Dirichlet , pi . -, Neumann pi , pi pi . pi . 6.5: pi Fourier cos
() [0, ].
pi.
cos()
==1
sin(
)
pi (9.1.20) ( 9) pi pi pi = 1 pipi
=
{0, = pi4
21 , =
(6.4.12)
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6.4. Fourier . 117
6.3:) cos
(),
) pi pi pi [,] cos ( ) , [0, ],) Fourier cos
(), [0, ]
Fourier cos() () (6.3)
pipi (), (5.2) (5) cos(3 ) pi pi (pi pi pi pi ), pipi (), (5.5) (5) ( sin( )) pi pi Fourier, Fourier .
pi pi pi - pi cos pi pi pi sin pi pi - pi sin pi pi cos ; pi pi pi pi !! pi pipi pi . pi [0, ] pi pi . pi pi pi !
pi pipi pi pi pipi pi , pi pi pi [,] pi pi . cos ( ) pi () (6.3), () pi pi pi pi cos
(), [0, ] [,]. pi ,
pi pi pi Fourier. pi pi - (5.2.93), pi cos
( ) sin
( )
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118 6. Fourier
[,] [0, ]. pi pi, pi
6.2. Fourier cos(3 )pi [,], .
6.4.3 Fourier.
pi Fourier, - pi pi pi (6.4.4).
, pi pi pi pi pipi pi [,] . (6.4.2) pi pi pi pi pi pi pi.pi 6.4.5 ( Fourier). (), [,] . Fourier () () [,] () () = ().
, pipi pi [0, ] [,].
pi 6.4.6 ( Fourier). (), [0, ] . Fourier () () [0, ] () .
pi 6.4.7 ( Fourier). (), [0, ] . Fourier () () [0, ] () (0) = 0 () = 0.
6.4.4 Fourier .
pi Fourier . pi- pi pipi, Fourier pi , pi pipi . pi pi pipi pi pi Fourier. 6.6 (pi pi Fourier): Fourier pi (6.3) () =, [0, ] pi pi pi
=4
pi.
1
sin()
pi , pi pi 4
pi.
cos()
pi !! pipi .
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6.4. Fourier . 119
6.7 (pi pi Fourier): Fourier () = , [0, ].
=2
=1
(1)+1 sin()
(6.4.13)
( pi pi.) , pi - dd = 1 pipi pi pi pi 1 Fourier. , pi pi
2=1
(1)+1 cos()
(6.4.14)
pi , 1 pi (6.4) pi 1 (pipi = 1 pi.)
pi pipi pi pi pi pi- pi pi, pi pi , . , pi pi pi pi pipi Fourier. (6.3.9) pi pi Fourier pi - pi , pi pi Fourier . -, pipi pi, pi (6.3.10), Fourier . pi pi Fourier .
Fourier. pi pi - pi pi (6.3.9) Fourier pi pi pi-. [,] pi pi Fourier pi .
pi pipi (6.3.9) pi (), [,] , pipi () = () (pi pi pi pi pi.). pipi 6.4.8 ( Fourier). (), [,] pi , , Fourier pi pi pi () = ().
pipi pi .
Fourier. pipi pi pi pi (6.3.10) - pi pi Fourier. pipi pi pipi 6.4.9 ( Fourier). (), [0, ] pi , , Fourier pi pi pi .
pipi pi . pipi pi pi .
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120 6. Fourier
Fourier. pipi pipi pi pi (6.3.10) pi pi Fourier. pipi pi, , pi
pi 6.4.10 ( Fourier). (), [0, ] pi , , Fourier pi pi pi (0) = 0 () = 0.
pipi pi .
. pi pi pi pi pi pi pi: pi pi Fourier !
pipi pi pi
pi 6.4.11 ( - Fourier). (), [0, ] pi , , . , pi Fourier
() =
=1
sin()
pi, , pi
=1
[() (0)] +
=1
[
+
2
((1)() (0))
]cos()
(6.4.15)
6.4.5 Fourier.
Fourier pi pi (6.3.3.ii) pi pi .
6.4.6 Gibbs.
pi pi 0 . pipi Fourier pi pipi pi 0 , pipi . pi pi , pi pi Gibbs pi pi pi pi , pi pi pi pi pi !
pi pi pipi 9% pipi . , pi, () = 100 pi pi 200 = 0, pi pi 100 100 = 0 pi pipi 9% 200 18.
Gibbs pi pipi pi- .
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6.5. Fourier. 121
6.4: Gibbs S16 .
6.8 (pi [16]):
() =
{12 0 < <
12 < < 0 pi pi pi , Fourier pi pi pi
pi=1
2
sin()
pi (6.3.7) (6.3.11) pi pi S16 pi (6.4).
6.5 Fourier.
. pi
d2
d2+ = 0
Dirichlet, Neumann, pi pi - pi pi :
sin(
), / cos
()
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122 6. Fourier
pi pi, pi pi pi .
, pipi pi pi (5.1.1) pi pi pi pi pi pi pi pi pi Dirichlet, Neumann, - , pi pi . , pi pi pi - pi pi pi pi pi !
pi
d2
d2+ = 0, < <
pi pi (5.1.1).
pi pi pi pipi pipipi (5.1.1) pi , pi pi , pi , , Fourier. pi pi , Robin, pi pi =
(
)2 pi pi .
- pi ; pi pi pi - pi pi pi pi Fourier; pi pipi pi , pi, pi . pi pi Sturm-Liouville.
pi .
: (), () [, ]. , pi ,
, =
()() (6.5.1)
pi pi. , - , (), pi | |2
| |2 = (), () (6.5.2) , pi
, =
()() = 0 (6.5.3)
pi pi .
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6.5. Fourier. 123
6.5.1 .
L = d2d2
pi pi pi.,
d2
d2+ = 0, < < (6.5.4)
(6.5.5)
pi , pi pi-. pi = L[] = d2
d2. ,
pi 1() 2(). pipi
L[1 ]() = 1() = 11()
L[2 ]() = 2() = 22()
(6.5.6)
1 = 2 pi , pi . pi pi pi
L[1 ]()2() + L[2 ]()1() =[
1()2() +
2()1()
] (6.5.7) pi
(L[1 ]()2() + L[2 ]()1()) =
[ 1()2() +
2()1()
] =
=[
1()2() +
2()1()
]
G2 (6.5.8)
G2 Green. , Green pipi , L. , , L. , , pi pi pi G2 !
(L[]()() + L[]()()) =
[()() + ()()] ==[()() + ()()]
G2 (6.5.9)
pi Dirichlet, Neumann,, Robin .
Dirichlet: 1() = 1() = 2() = 2() = 0 G2 = 0
Neumann: 1() =
1() =
2() =
2() = 0 G2 = 0.
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124 6. Fourier
: 1() = 2() 2() = 2() G2 = 0.
Robin: Robin
+ = 0, = 1, 2. pi, () = ()
() = (). pi Green
[ 1()2() +
2()1()
]
=
= 1()2() +
2()1() +
1()2() 2()1() =
= 1()2() 2()1() 1()2() + 2()1() = 0 (6.5.10) Robin G2 = 0.
, G2 . pi, () = (), () = () pi pipi pi G2 .
pi pi pi ; pi ;
[ 1()2() +
2()1()
] =
[11()2() 22()1()] =
=
(1 2)2()1() =
= (1 2)
2()1() (6.5.11)
pi pi pi (6.5.6). pi pi 1 = 2 pi pipi
(1 2)
2()1() = G2 (6.5.12)
, pipi pi G2 = 0 ( pi Dirichlet, Neumann, , Robin pi G2 = 0) pipi pipi
2()1() = 0 (6.5.13)
(1 2) = 0. - Green pipi pipi pi ! pi , pi , pi pi Green pi ! pi .
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6.5. Fourier. 125
: pi ,
1() + 1() + 1() + 1 () = 0
2() + 2() + 2() + 2 () = 0
(6.5.14)
pi pi pi [
()() ()()]
= 0 (6.5.15)
pipi () () pi pi (6.5.14). pi pi pi ! pi ! pi !
pipi pi pi pi pi . 6.5.1 ( .). pipi , pipi (6.5.4) (6.5.14), pi . pi, pipi pi pi pi pi pi pi.
pi. () pi pi pi (). ,
() =
() (6.5.16)
. pi ,
(), () =
(), ()
=
=
, =
= , = | |2 (6.5.17) pi . ,
=(), ()
| |2(6.5.18)
pi pi ().
pi,pi . pi , , 1 = 2 . pipi pi pi pi . , , pi pi Gram-Schmidt pi pi pi pi , pipi .
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126 6. Fourier
. pi, pi pi- pi ( pi pi pi ) pi pi pi pi pipi. pi pi pi pi - ; pi pi pi pi (6.5.1).
6.5.2 ( ). pi pi pipi pi pi, pi.
pi. pi
pi pi pi pi . pi pi pi (6.5.1).
6.5.3 ( ).
() ()
0 (6.5.19)
pi , (), pi pi , pi .
pi. pi
6.3. pipi Robin pi pipi .
, pi , pi Robin, pi pi pi. pi ; e . Dirichlet pi pi pi . pi pi . Neumann pipi pi . pi pi pi , pi . , pi , Neumann pi pi pi pi pi pi pi pi . , Robin pi pi . , pipi pi pi pi pi pi pi . pi pi e < 0.
pi pi pi pi pi : 1) pi pi pi
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6.5. Fourier. 127
pi pi 2) pi pi pi ; pi (6.5.1) pi, pi pi pi: pi pi pipi , .
pi pi pi Sturm-Liouville pi pi, pi pi ,pi pi pipi, pi pi .
pi pi pi pi pi , pi Fourier , pi Fourier pi .
6.5.2 Fourier.
pi pi () - . pi pi :
d2
d2+ = 0, 0 < <
pi (). pipi pi pi
() =
(), (6.5.20)
pi , f pi pi
f =, ,
=
()()
2
()
6.5.4 ( ). Fourier
f()
() [, ] 1. (), () () pi 2. () pi .
pi. pi.
Fourier pi pi ()
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128 6. Fourier
pipi pi pi Fourier 2 . 6.5.5 (2 ). Fourier
f()
2 () (, )
|()|2
pipi.
pi. pi.
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9
9.1 . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.1 .
.
sin = cos(
2 ) (9.1.1)
cos = sin(
2 ) (9.1.2)
sin( ) = sin cos cos sin (9.1.3)cos( ) = cos cos sin sin (9.1.4)
sin 2 = sin cos (9.1.5)cos 2 = cos2 sin2 = 2 cos2 1 = 1 2 sin2 (9.1.6)
sin cos =1
2[sin( + ) + sin( )] (9.1.7)
cos sin =1
2[sin( + ) sin( )] (9.1.8)
sin sin =1
2[cos( ) cos( + )] (9.1.9)
cos cos =1
2[cos( + ) + cos( )] (9.1.10)
sin2 sin2 = sin( + ) sin( ) (9.1.11)cos2 cos2 = sin( + ) sin( ) (9.1.12)cos2 sin2 = cos( + ) cos( ) (9.1.13)
152
-
9.1. . 153
sin() sin() =
sin[( )]2( )
sin[( + )]
2( + )+ , || = || (9.1.14)
sin2() =
2 1
4sin(2) + =
2 1
2sin() cos() + (9.1.15)
cos() cos() =
sin[( )]2( ) +
sin[( + )]
2( + )+ , || = || (9.1.16)
cos2() =
2+
1
4sin(2) + =
2+
1
2sin() cos() + (9.1.17)
sin() cos() = cos[( )]
2( ) cos[( + )]
2( + )+ , || = || (9.1.18)
sin() cos() = 12
cos2() + (9.1.19)
pipi pi = , =
0
sin(
) cos(
) = cos[( )]
2( ) cos[( + )]
2( + )+
2( )+
+
2( + )=
{0, = 2 (
22 ), = pi
(9.1.20)
, = + = = pi + = pi
: 7 2014
-
[1] ., . (1994). . .[2] . (1991). . .[3] pi (2009). . , http://eclass.uth.gr/MHXC109.
, pi .
[4] pi (2011). . -, http://www.mie.uth.gr/n_ekp_yliko.asp?id=33. , 4 - pi .
[5] Asmar N. H. (2005). Partial Differential Equations, With Fourier Series and Boundary ValueProblems. NJ: Pearson-Prentice Hall.
[6] Courant R. and H. D. (1962). Methods of Mathematical Physics, Vol . New York: Wiley.[7] Evans L.C. (1998). Partial Differential Equations. Providence: American Mathematical Society.[8] Folland G.B. (1992). Fourier Analysis and its Applications. Belmont, California: Wadsworth.[9] Freiling G. and V. Yourko (2008). Lectures on Differential Equations of Mathematical Physics, A
First Course. New York: Nova Science Publishers.[10] Haberman Richard (1987). Elementary Applied Partial Differential Equations, With Fourier
Series and Boundary Value Problems. NJ: Prentice-Hall, Inc .[11] Haberman Richard (2004). Applied Partial Differential Equations, With Fourier Series and
Boundary Value Problems (Fourth .). NJ: Pearson/Prentice-Hall.[12] Powers David L. (2006). Boundary Value Problems and Partial Differential Equations (Fifth .).Amsterdam: Elsevier.
[13] Renardy M. Rogers R.C. (2003). An Introduction to Partial Differential Equations. New York:Springer.
[14] Snider A.D. (1999). Partial Differential Equations, Sources and Solutions. NJ: Prentice Hall.[15] Spivak Michael (1994). pi,
. : pi .[16] Strauss W.A. (2008). Partial Differential Equations, An Introduction. Hoboken, NJ: Wiley.[17] Pinchover Y. and Rubistein J. (2005). An Introduction to Differential Equations. Cambridge:Cambridge University Press.
154
Seir'ec [Fourier]Seir'ec [Fourier].Hmitonik'ec, Sunhmitonik'ec, kai Pl'hreic Seir'ec [Fourier].
H Seir'a [Fourier] miac Periodik'hc Sun'arthshc.Periodik'ec Sunart'hseic.'Artiec kai Peritt'ec Sunart'hseicPl'hreic Seir'ec, Peritt'othta kai Arti'othta.H Seir'a [Fourier] M'iac Peritt'hc Sun'arthshc.H Seir'a [Fourier] M'iac 'Artiac Sun'arthshc.
Jewr'hmata S'ugklishc.E'idh S'ugklishc Seir'wnTo Je'wrhma S'ugklishc.Par'agwgoi kai Oloklhr'wmata Seir'wn [Fourier].Par'agwgoc Seir'ac [Fourier].Olokl'hrwma Seir'ac [Fourier].
Seir'ec [Fourier] se Diast'hmata.Periodik'ec Epekt'aseic Sunart'hsewn.Oi Seir'ec [Fourier] twn Periodik'wn Epekt'asewn.H seir'a [Fourier] thc Periodik'hc Ep'ektashc.H Seir'a [Fourier] thc 'Artiac Periodik'hc Ep'ektashc.H Seir'a [Fourier] thc Peritt'hc Periodik'hc Ep'ektashc.Sunhmitonik'ec kai Hmitonik'ec Seir'ec [Fourier].To Je'wrhma S'ugklishc.Sqed'iash Seir'wn [Fourier].
H Sun'eqeia thc Seir'ac [Fourier].Parag'wgish Seir'wn [Fourier] se Diast'hmata.Olokl'hrwsh Seir'ac [Fourier].To fain'omeno [Gibbs].
M'ejodoc Qwrismo'u Metablht'wn kai Genikeum'enec Seir'ec [Fourier].Orjogwni'othta kai SS. S'ugklish Genikeum'enwn Seir'wn [Fourier].
Par'arthmaP'inakec Trigwnometrik'wn Sunart'hsewn.