Математический анализ — Ряды Интегралы Фурье -…


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Фɭрье
Фɭрье
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Фɭрье
,
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,
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ɮɭнкций
ɫɬɭɞенɬоɜ
ɮакɭльɬеɬа
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Пеɬерɛɭрɝɫкоɝо
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ɭниɜерɫиɬеɬа


.
Пеɬерɛɭрɝ

.
fxbnxx
()sin,(,)
bFxnxdxfxnxdx
()sin()sin
(8),
0


[,]
,

(8)
[,]

(8)
[,)
fxx
[,]
gxx
[,)

()()
xSx
(
()()
y
yfx
ɮɭнкции
π−ππ
ɮɭнкции

yfx
ygx
ɮɭнкции
1.
,
()()
()()
(3)
[,]
2.
(4)
Aab
(3)



[,]
−ππ



()(),[,)
()();()()


[,]
fxAanx
()cos
AFxdxfxdx
()()
aFxnxdxfxnxdxn
===
()cos()cos,,,
()()
[,]

,
0

()(),[,)

(,)
yfx
ygx
ɮɭнкции
gagx
()lim()
()()
gxga


fxxa
gxxa
(),[,],
(),[,).
еɫли
еɫли
(1)

[,]
=−π
FxAanxbnx
()(cossin)
=++

(,)
fxAanxbnx
()(cossin)
=++
(3)
()()
(,)
(3),


gafa
()()
0
2
(


()()
(3)

AFxdxgxdxfxdx
aFxnxdxgxnxdxfxnxdxn
bFxnxdxgxnxdxfxnxdxn
==+
==+
==+
∫∫∫
∫∫∫
∫∫∫
()()();
()cos()cos()cos,,,;
()sin()sin()sin,,,.
(4)
ygx
yfx
ɮɭнкции
−ππ
ɮɭнкции
−ππ
ɮɭнкции

ππππ
22222
222
4126
=++++++
⇒=−=
2222
+++=
2222

2222
+++
4624
22222222
+++=++
+++
=⋅=
KKKK
6248
2222
222
+=−=
πππ
[,]
−

[,)



xxxnx
=⋅−+−+−⋅+
sinsinsin

()()

ɮɭнкции
−ππ
ɮɭнкции
fxx
[,]

,
()()

(
Axdx
axnxdx
xdnx
xnxxnxdx
===−
∫∫∫
222
πππ
cossinsinsin
==−
==−⋅
444
xdnx
xnxnxdx
nnn
πππ
coscoscos
(,,)
xxxnx
222
=−⋅−+−+−+
coscoscos
cos
[,]
()()

Afxdxafxnxdx
();()cos
fxR
()[,]
∈−ππ


fxdx
()0,
(4)
afxnxdxn
=⋅=
()cos0,,,,
(5)
bfxnxdxfxnxdxn
=⋅==
()sin()sin,,,.
(6)
fxR
()[,]
∈−π
fxbnx
()~sin
bfxnxdx
()sin
[,]

()()
−=−
bxnxdx
xdnx
xnxnxdx
==−⋅=−−
∫∫∫
222
πππ
sincoscoscos
=−=−⋅−=−⋅
cos()()
∈∈−
(,)

()()

()()
1
2
−π−ππ
ɮɭнкции
fxR
()[,]
∈−ππ

fxdxfxdx
()(),
(1)
afxnxdxfxnxdxn
=⋅==
()cos()cos,,,,
(2)
bfxnxdxn
=⋅==
012
()sin,,,.
(3)
fxR
()[,]
∈−π
fxAanx
()~cos
()()
fxtfx
()()
fxtfx


fxfx
()()



()()
−++−
()()
ftf
()()
ftf
(,)

()()
()()
fxfx
()()
−++
,[,);
,(,].
еɫли
еɫли
еɫли
fxR
()[,]
∈−ππ
Afxdxfxdxfxdxdxdx
==+
=⋅+⋅
−−−
∫∫∫∫∫
πππ
()()()
afxnxdxnxdxnxdxn
==⋅+⋅
∫∫∫
01012
()coscoscos,,,
bfxnxdxnxdxnxdx
==⋅+⋅
∫∫∫
()sinsinsin
=−=
(,)(,)
xxx
sinsinsin
sin()
=+++++
1
2
2
1
3
3
5
5
−ππ
ɮɭнкции
gxxgx
fxxfx
()()()()
()()
()()
gxxgx
fxxfx
()()
I





ggtftf
()lim()lim()()
−==
→−→−
ggtgtgtft
tttt
()lim()lim()lim()lim()
+==
→+→+→+→+
0000
,
()()
()()
gtg
gtf
→+→+
+−+
−+−−
()()
ftf
()()
()()
gtg
ftf
→+→+
−−−
,






ggff
()()()()
πππ
−++

()()
1.
()()
[,]
2.

()()
−∞+∞
(,)
3.
fxR
()[,]

(,)


[,]


fxR
[,]
−ππ
fxx
(),(,);
()()
еɫли
еɫли
2
(

(,)
−ππ


gxfxx
gxgx
()(),[,);
()().
еɫли
2
yfx
−ππ
ɮɭнкции
y
ygx
π−ππ
ɮɭнкции
gxR




,
aggtntdtafftntdtn
()()cos;()()cos,,,
===

()()
[,)
−ππ
agaf
()()
(,)

()()
()(,)

xxux
(7)
0
fxR






I,
fxtfxtfx
()()()
++−−
0


()()()
()()()()
fxtfxtfx
fxtfx
fxtfx
→+→+
++−−
fxfx
()()
fxR
()()
fxtfx
()()
fxtfx


fxfx
()()
−++


fxtfxtfxfx
()()()()
++−−+−

0


()()()()
fxtfxtfxfx
()()()()
fxtfx
fxtfx
+−+
−−−
2





2.
fxtfxtA
()()
++−−



fxtfxtA
fxtfxtA
1221
()()()()
++−−
++−−
fxtfxtA
fxtfxtA
1221
()()()()

fxR





I
fxtfxtfx
()()()
++−−

.)
()()()
fxtfxtfx
()()()()
()()
fxtfx
fxtfx
fxfx
°. Пɭɫɬь ɮɭнкция fxR






Sxfxtfxt
()()()
sin()
=++−⋅
1
2
2
sin()

(2)


SxAfxtfxtA
()()()
sin()
−=++−−⋅
222
(3)




()()
sin
sin()
fxtfxtA
ntdt
++−−
222
210

(4)
fxtfxtA
()()
sin
++−−
222
(5)
fxtfxtA
()()
++−−
222
fxtfxtA
()()
++−−
(6)
fxR


fxtfxtA
()()
++−−





fxtfxt
++−
(4)
fxR






[,]




.
Sfx
(,)
Sgx
(,)



Sfxfxtfxt
dtx
(,)(
sin()
=++−⋅
Sgxgxtgxt
dtx
(,)(
sin()
=++−⋅
,
t
a
0
2

[,]
fxtfxt
sin()
++−⋅
=++−⋅
gxtgxt
sin()
sin
SfxSgxxx
nnnn
(,)(,)()()
SfxSgx
(,)(,)
,
[,]

lim()sin
tztdt
→+∞
()sin
tztdt
lim()
→+∞
2.
lim()cos
tztdt
→+∞
ftR
()[,]
aftntdtbftntdt
=→=→
()cos;()sin
[,]
−ππ

fxR

Sxfxtfxt
()(
sin()
=++−⋅

Sxfxtfxt
dtx
()(
sin()
=++−⋅
xfxtfxt
()(
sin()
=++−⋅
()sin
tztdt
⋅≤⋅

+⋅
2



2
M

lim()
→+∞


(,]
[,]
tdt
lim()sin
tztdt
→+∞

tdt
lim()()
tdttdt
()()
tdttdt
0
tdt
0


0

:
tdt
Jztztdttztdttztdt
()()sin()sin()sin
==+
∫∫∫
ψψψ

ψψψ
ααα
()sin()sin()
tztdttztdttdt
aaa
∫∫∫
≤⋅≤
lim()sin
tztdt
→+∞

[,]
attttb
012
kkk
0
1

[,]

[,]
[,]
Jztztdt
()()sin
Jztztdtttztdttztdt
()()sin()()sin()sin
==−+
+++
ψψψψ
111
ttt
[,]
()()
ψψψψ
()()sin()()sin()
ttztdtttztdttt
kkk
−≤−
ψψω
()()sin()
ttztdttt
kkk
−≤−
Jztztdt
()()sin
+⋅

[,]
()[,]
tRab
sup()
[,]
coscos
ztdt
ztzt
Jfxt
sin()
Jfxt
dtJfxt
sin()
sin()
Sxfxtfxt
()(
sin()
=++−⋅
fxR

fxR


sin()
~
2
2
SxAakxbkx
nkk
()(cossin)
=++
1
Adt
=⋅=
aktdtkbktdtk
=⋅===⋅==
1012
1012
cos(,,);sin(,,)
2
2
sin()
sin
()[,]
tRab
Sxft
()()
sin()
(;)
−∞+∞
fxR


Sxft
()()
sin()
2
2
Sxfxu
()()
2
2

(
−−−−=
()2

3
[,]
−−−

Sxfxu
()()
sin
2
2


Sxfxt
()()
sin()
∫∫∫
2
2
0
2
1
2
0
2
J

⋅=+−
sinsinsinsinsinsin
⇒⋅=
sinsinsinsin
012
,,,
n
sin
(1)
012
,,,


.

limcoscoscos
ααα
++++
=+=
απβ
βαπ
замена
еɫли
()sin
()sin
βββ
nnn
fxR
()[,]




SxAakxbkx
nkk
()(cossin)
=++
Aab
Aftdtaftktdtbftktdt
===
−−−
∫∫∫
πππ
();()cos;()sin
Sxftdtftktkxktkxdt
()()()(coscossinsin)
=++⇒
⇒=+−
Sxftktxdt
()()cos()
+−=
cos()
sin()
ktx
,
[,]
Aab
fxR
()[,]
Afxdx
afxnxdx
()cos
bfxnxdx
()sin
(6)
5.
[,]
fxR
()[,]
∈−ππ
[,]


Aakxbkx
(cossin)
fxAakxbkx
()~(cossin)


++++=
coscoscos
ααα
=++++
coscoscos
ααα
sin
⋅=++++
sinsinsincossincossincos
ααα
2sincossin()sin()
=+−
1
2
[,]
[,]
Aaxbxaxbx
anxbnx
+++
(cossin)(cossin)
(cossin)
1122
(6)
4.
[,]
[,]
−ππ
Aabab
,,,,,
1122
[,]
fxAaxbxaxbx
anxbnx
()(cossin)(cossin)
(cossin),
+++
1122
(7)
(7),
[,]
−ππ
(7)
(7)
(7)
[,]
fxdxAdxfxdxA
()()
−−−
∫∫∫
=+⇒=⋅⇒

Afxdx
(7)
(7)
[,]
−ππ

(T)
[,]
fxnxdxanxdxa
()coscos
=⋅=⋅⇒

afxnxdxn
()cos(,,)
(7)

bfxnxdxn
()sin(,,)


[,]
−ππ
sincossin()sin()
pxqxpqxpqx
⋅=++−
cossin
nxdxnxdx
nxdx
dxdxnxdx
−−−−
∫∫∫∫
=+=
полемме
cos
nxdx
dxdxnxdx
−−−−
∫∫∫∫
=−=
полемме
3.

()()
∈−∞+∞
(,)

()()
xdxxdx
ϕϕϕϕ
()()()()
xdxxdxxdxxdx
∫∫∫∫
=++
1
Juduudu
=+=
ϕπϕ
()()
()()
xdxJxdx

(3)
(4)
(2)
ftztdt
()sin
()()sin
zftztdt
fxzzxdz
()()sin
(5)
(6)
(3)

(4)
ftztdte
()cos
fxexzdz
()cos
221
z
2

[,)
()0
0
I
2



lim()
→+∞
IIe
[,)
(,)
fxzxdzftztdt
()cos()cos
(1)
fxzxdzftztdt
()sin()sin
(1)
ftztdt
()cos
()()cos
zftztdt
fxzzxdz
()()cos

(3),
(4),
y
2
ɮɭнкции



(,)

(
),










IIe
IIe
11,
∈+∞
[,)
IxItztdtzxdz
()()coscos
zxdzeztdt
()coscos
eztdt
zxdz
cos
0
2

zzx
dzex
101
при
при
fxfx
azzxdz
()()
()cos
−++
azftztdtztdt
()()coscos
sinsin
===⋅=⋅
2222
ππππ

zxdz
при
при
при
(5)
(5)
fxMzzxdz
()()sin()
y
ɮɭнкции
(,)

Ftdt
FtFtF
FtF
ftf
()()()()()()()
+−−
∫∫∫
(III
(1)
1.
fte
eeztdtzxdzx
coscos,
eeztdtzxdzx
sinsin,
eztdt
eztdt
=−=−−=−
=+∞
cos()cossin
ztdeeztzeztdtz
==−=−+=
=+∞
∫∫∫
eztdtztdeeztzeztdtz
ttt
sinsin()sincos
1
1
z
z
z
z
,
;


:
dzx
cos
zzx
1
0

dzex

Ftdtftdt
()()
+∞+∞
FxtFxtFx
fxtfxtfx
()()()()()()
++−−
++−−


(III
Fxazzxdz
()()cos
azFtztdtftztdt
()()cos()cos
∞+∞
()()
fxftztdtzxdz
()()coscos
(),
().
ftt
ftt
−−
ɞля
ɞля

()()()()
Ftdtftdtftdtftdt
∫∫∫∫
=−+=
()()()
FxtFxtFx
fxtfxtfx
++−−
++−−

(III

(III
ftt
ftt
(),
().
ɞля
ɞля
(III)
Mzazbz
()()();
sin;
cos
=+==
azzxbzzxMzzxzx
()cos()sin()(sincoscossin)
Mzzx
()sin()
fxMzzxdz
()()sin()
[,)

[,)
ftdt

fxtfxtfx
()()()
++−−
fxftztdtzxdz
()()coscos
fxftztdtzxdz
()()sinsin
ftf
()()
(2)
-


ftt
ftt
(),
().
ɞля
ɞля
fxdzftztzxztzxdt
()()(coscossinsin)
ftztdtaz
()cos()
ftztdtbz
()sin()
fxazzxbzzxdz
()()cos()sin

azftztdt
()()cos
fxazzxdz
()()cos
2)

bzftztdt
()()sin
fxbzzxdz
()()sin
(III)
(III)
0
0

(1)
(2),
0





ftztxdt
()sin()

ftztxft
()sin()()
ftdt
()sin()


ftztxdt
()sin()

dzftztxdt
()sin()

dzftztxdt
()sin()
v.p.()sin()
dzftztxdt
2

fxdzftztxiztxdt
()v.p.()cos()sin()
=⋅−+−=
2
=⋅⋅
v.p.()
dzftedt
iztx
cossin
fxdzftedt
iztx
()()
2



fxdzftztxdt
()()cos()
cos()coscossinsin
zxz
−=+

(,)

fxdx




fxtfxtfx
()()()
++−−
fxdzftztxdt
()()cos()





fxtfxtfxfx
()()()()
++−−+−−
fxfx
dzftztxdt
()()
()cos()
−++
1



(,)
ftztxdt
()cos()

dzftztxdtdzftztxdt
()cos()()cos()
−=−
2


fxdzftztxdt
()()cos()
2
Fzdzdzftztxdtftztxdzdt
aaa
()()cos()()cos()
000
=−=−
dzftztxdtft
atx
()cos()()
sin()
(17),
∫∫∫




dzftztxdtfxtfxt
()cos()()()
−=++−
fxtfxtt
()()()
111
[,)

(,)
()()()
tdtfxtfxtdt
111
=++−
fxtfxtdtfxtdtfxtdt
()()()();
++−≤++−+∞
∫∫∫
111
1
ɫхоɞиɬɫяɫхоɞиɬɫя
()()()()()
fxtfxtfx
++−−

dzftztxdtfxfx
()cos()()()
−→⋅=⋅
→+∞
0
()()
atdt
tdt
+∞+∞+∞
∫∫∫
≤
sin
→+∞
0
AAA
max{,}
()()
−⋅
()()
→+∞

(,)
−∞+∞
fxdx
(,)
fxtfxtfx
()()()
++−−
fxdzftztxdt
()()cos()
()cos()


(16)).
ftztxft
()cos()()

ftdt
ftztxdtFz
()cos()()


Jat
dtt
()()
=+=
∫∫∫
()()
sin()
atdt
dtt
sinsin
MaM
t



atdtt
()()
sin()
sin
∫∫∫
0
==+
∫∫∫
sinsinsin
()()
⋅=⋅+⋅
t
t

atdtt
()()
()()
sin()
∫∫∫
∞+∞
ϕϕϕϕϕϕ
()()()()()()
∫∫∫
000

1

()()

()()
atdt
→+∞
tdt
()()
lim()
→+∞
⋅=⋅
Jat
()()

0


Jat
()()
1


lim()




1

t
tdt
tdt

tdt
0
tdt
Fzftztxdt
()()cos()

(7)

(7),
ftztxft
()cos()()
ftdt
n
0123
=====
;;;;;;
Fzft
txdt
()()cos()
()()
zzz
nnn
=−=
(8)
fxFzz
()()
(9),


Fzdz



fxFzdz
()()
(10)

fxFzdzfxdzftztxdt
()()()()cos()
=⇒=−

[,)

ftdt
ftdt
≤≤=
()()
ftdt

ftdt
0
→+∞


fxa
()cossin
()()coscossinsin
≈⋅+
ππππ

coscossinsincos()
+=−
txdt
()()cos()
txdt
()cos()
txft
()cos()()
ftdt

txdt
()cos()

txdt
()cos()
txdt
()()cos()


−++−+
coscossin
=⋅+
sinsin
sinsinsin
123
,,,
fxC
()[,]
()()
[,]
cos
cos
3
3
1
2
3
2
3
−−−−
4321
1234567
ɮɭнкции

(,)
−∞+∞
(,)
∞+∞

fxdx
−
[,]


fxAa
()cossin
=++
ftdta
dtb
dtn
n
n
====
∫∫∫
(),()cos,()sin,,,
=++−
()cos
=++−
∫∫∫
dxx
coscos()cos
πππ
=+++
sincossin
+−−
sinsincos
=−+−+−
2222
sincossin
−−+−+
2222
sinsinsincos
coscoscos
=⋅−
=−⋅
2222
cos
,,,
()sin
=++−
∫∫∫
dxx
sinsin()sin
πππ
=−+
cossincos
−−−+
coscossin
=−+−+−
cossincoscos
cos()cos()
cos()cos()

cos()cos()
2121
sin()
sin()
=−⋅⋅
2222
πππ

cos
2
2
1
8
[,]
[,]
Aab
[,]
[,]
−≤≤
еɫли
еɫли
еɫли
112
323

3
2
fxdxxdxdxxdx
==+⋅+−
∫∫∫∫
()()
123
ɮɭнкции
dzaf
nzdzbf
nzdz
−−−
∫∫∫
ππππππ
;cos;sin
(1),
=−π
flfl
=−=
()()
x

[,]
[,]
(,)
fxAa
()cossin
=++
Aab
x

fxdxa
dxb
n
n
===
∫∫∫
();()cos;()sin


=−π
flfl
()()

()()
1.
()cos
[,],
−=−
()cos()cos()
−=−

123
,,,
dxxxnxdx
===
()sincossin
=⋅++−=
2121
xnxnxdx
sin()sin()
ɮɭнкции
ɮɭнкции
fxRll
()[,]

[,]
[,]
[,]
−ππ
fxRll
()[,]
[,]
[,]
[,]
−ππ
[,]

[,]
−ππ
[,]

[,]
(,)
Aanzbnz
=++
(cossin)
()()

2222
fxdxAab
()()
=++
2222
gxdxPpq
()()
=++
222
fxgxdxAPapbq
nnnn
()()()()()
+=+++++
fxgxdxAPapbq
nnnn
()()()
=++
Ppq
fxgxdxAgxdxagxnxdxbgxnxdx
()()()()cos()sin
−−−−
∫∫∫∫
=++
fxAanxbnx
()~(cossin)
(17)
[,]
−ππ
[,][,]
∈−ππ
xlm
xlm
[,],
[,]\[,].
при
при
fxdxAdxanxdxbnxdx
()cossin
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=++
(17)
[,]
−ππ
xaa
(,)
0121
,,,,
(12)
fxdxaa
iii
()()
≤−⋅
fxdxaa
iii
()()
≤−⋅⋅=
+⋅
ρρε
+⋅
n
6

fxx
[,]
[,]
xxxnx
222
=−−+−+−+
coscoscos
n
n
4
2222
fxdxAab
()()
=++
ππππ
xdx
nnn
nnn
∑∑∑
=⋅+⇒=+⇒=
fxR
()[,]
∈−ππ
Aab

gxR
()[,]
∈−ππ
Ppq
2
fxgxdxAPapbq
nnnn
()()()
=++

()()
3
n
3

n


fxR
()[,]
∈−ππ

[,]
−ππ
aaaa
012
=−
,
iii

[,]

[,]
[,]

xais
ɜɭзлах
xaais
,,,,,,();
,(,),,,,.
∈=−
0012
011

n
fxfxfx
()()()
fxdx
fxdxfxdx
()()
xaa
(,)
fxfxf
()()
fxdxaa
iii
()()
[,]



fxM
[,]
ais
axais
линейнаɞля
axa
иɞля
axa
iii
iiii
()(,,,,);
()(,,,,);
().
=+≤≤−=−
−≤≤+−≤≤
0012
0121
2

yfx
[,]

,
()()
−=+=
1.)
n
,
fxM
[,]


fxfxfx
()()()
fxdx
fxdxfxdxfxdxfxdx
()()()()
==+
[,]
fxfx
()()
0

fxfxfxfxfxfxM
()()()()()()
=−⇒≤+≤
fxdxM
fxdxM
fxdxMs
216
ɮɭнкции
yfx
()()
ABAB
−≤+
222
fxSxfxSxfxSx
nnn
()()()()()()
−≤⋅−+−


nnn
2()
nkk
fxdxAabfxdx
=−++
2222
()()()
[,]


()()

TxPpkxqkx
mkk
()(cossin)
=++

fxTx
()()
[,]

fxTxdx
()()

mmm
fxSxdxfxTxdx
=−≤−≤
()()()()




n





[,]
−ππ
−==
aaaa
012
[,]
(,)
xaa
(,)
fxc
0121
,,,
). (

ɫɬɭпенчаɬой
ɮɭнкции
TxSx
()()

222
Aabfxdx
++≤
()()
222
Aab
(6)
(5)
222
Aabfxdx
++≤
()()

222
Aabfxdx
++=
()()

n
fxR
()[,]
−ππ
222
Aabfxdx
++=
()()


012
≥≥≥≥≥

fxfxfx
()()()
[,]
SxSxSx
nnn
(),(),()
fxfxfx
(),(),()
nnn

fxdx
SxSxSx
nnn
()()()
fxSxfxSxfxSx
nnn
()()()()()()
−=−+−
=++
22222
Pdxpkxqkxdx
(cossin)
122
,cos,sin,cos,sin,,cos,sin
nxnx
[,]
−ππ
2222
TxdxPpq
()()
=++
rfxfxTxTxdx
=−+=
()()()()
=+++−++
222
2222
fxdxPpqPApaqb
kkkk
()()()
⇒=+−+−+−−
rfxdxPApaqb
nkkkk
2222
()()()()
−++
222
Aab
(3)
Ppq


,,,


,


fxR
()[,]


SxAakxbkx
nkk
()(cossin)
=++
fxSxdx
()()


nkk
fxdxAab
=−++
2222
()()


QxQxQx
++++
QxQxQxPx
+++=
()()
Pxfx
[,]
(8).
(8)
[,]
fxQx
[,]
(8)
[,]


fxR
()[,]
−ππ
TxPpkxqkx
()(cossin)
=++
rfxTxdx
()()


Afxdxafxkxdxbfxkxdx
===
−−−
∫∫∫
πππ
();()cos;()sin
fxTxdxfxPpkxqkxdx
()()()(cossin)
=++
=⋅++⇒=++
APapbqfxTxdxPApaqb
kkkk
kkkk
()()()()
TxdxPpkxqkxdx
()(cossin)
=++
[,]
−ππ
Qycy

[,]
ayba
++−
()()
[,]
xaba
2
()()
[,]
(7),
fxc
xaba
()()
−−−
()()
Pxc
xaba
−−−


[,]
k

[,]

2
3
1

0

[,]
()()
Pxfx
−ε
()()
Pxfx
[,]
[,]
[,]


()()
Pxfx
[,]
(),
QxPx
(),
QxPxPx
221
(),
QxPxPx
332
(),
QxPxPx
nnn
KKKKKKKKK
KKKKKKKKK
TxPxpkxCkxqkxSkx
kmkm
()cos()sin()
−≤⋅−+⋅−
+⋅=⋅=
εεε
222
fxTx
()()

[,]
[,]
TxPx
2
fxPxfxTxTxPx
()()()()
−=−+−⇒
⇒−≤−+−+=
fxPxfxTxTxPx
()()()()
[,]



()()
−−+
()()
()()

()()
[,]
()()
+=−

[,]
gxPx

fxAxPxfxPxAx
()()
+−⇔−−
PxPxAx
[,]
fxPx
−ε
[,]
xaba
()()
(5)
[,]
[,]



ayba
2
()()
[,]
[,]
ayba
++−
()()
(,)
∞+∞
fxQx
()()
∈−∞+∞
(,)
[,]
0
Pxccxcx
=+++
[,]
fxPx
−ε
=−π

()()
()()
(,)
−∞+∞

TxPpkxqkx
()(cossin)
=++

:
fxTx
()()

pqM
cos
!!!
zzz
=−+−+
357
357
cos
!!!
zzz
=−+−+
246
246
(2)
(3)
[,]
Sxz
Czz
()sin;()cos
−−

()()()
PxPpCkxqSkx
kmkm
=++


[,]
123
,,,,
[,]

Ckxkx
Skxkx
()cos;()sin
−−







TxTxTxTx
123
(),(),(),,(),
Txfx
()()
∈−∞+∞
(,)

xfx
()()
(,)
A.


0

(,)
fxTx
()()

n





Txfx
()()
(,)
QxTx
QxTxTx
QxTxTx
QxTxTx
nnn
221
332
()(),
()()(),
()()(),
()()(),


QxQxQx
()()()
QxQxQxTx
()()()()
Txfx
()()
∈−∞+∞
(,)
(1)

0
n
const


2
n

fxfx
()()
++−

fxfx
()()



∈−∞+∞
(,)
()()()
Htfxtfxfxtfx
()()()()()
=+−+


0


δδε
(,)







σσε
fxfx
xfx
(()
()()
++−
=−
xfx
()()
ftR





,

xfx
()()
TxPpkxqkx
()(cossin)
=++
[][]
=+−++−−−⋅
2020
fxtfxfxtfx
()()()()
fxtfxfxtfxHt
()()()()()
2020
fxfx
()()
sin
++−
(4)




ftR

-

Mft
sup()
fxfx
()()
++−
2
0
2
0
2
0
2
3
t
dt
n
nt
t
dt
n
nt
t
dt
sin
sin
sin
sin
sin
.

,



)
,
fxfx
()()
++−
,
2
sin
t
2
222
sin
sinsinsin
≤⋅⋅−
⋅⋅⋅=
fxfx
()()
++−
ftR
Sxfxtfxt
dtp
()()()
sin()
,,,,
=++−
012
fxtfxt
ttntdt
()()
sinsinsin()
++−
+++−
321
=+++
sinsinsin()
321
2sin
2223221
Ttttttnt
sinsinsinsinsinsin()
=+++−
212
sincos
2sinsincos()cos()
2122446
sin(cos)(coscos)(coscos)
+−−=−=⇒=
cos()coscossin
222122
ntntntntT
(




fxtfxt
=++−⋅
()()
ftR

SxSxSxSx
0121
()()()()
====
SxSxSx
()()()
011
2
0
2



0
(3)
fxfx
()()
++−
fxfx
()()
⇒−≤
−+−++−
xlxlxl
⇒−≤
xlxlxl
xlxlxlA
−+−++
−+
n

max(,)




n



2
2
122
SSS
==→
1221
2121
SSS
n
1
2
3 (
ftR
Aakxbkx
(cossin)
SxASxAakxbkx
ikk
();()(cossin)
==++
SxSxSx
()()()
011
fxfx
()()

xfx
()()
xfx
()()


aaa
(1)
Saaa
lim()

aaa
Saaa
lim()
122


aaa
Saaa
SSS

n
(1)

2.

xxx
n

2

xxx
xlxlxl
+++
1212
()()()
⇒−≤
−+−
xlxlxlxlxl
mmn
121
..................................................... 3
......................................................................... 3
....................................................................................... 6
........................................................................... 10
........................................... 14
................................................ 20
................................................................................................ 23
.................................................................... 27
........................................................... 28
........................................................................................... 32
............................................................ 39
[,)
...... 42
............................... 45
............................................................................. 47
............................... 48
)..................................... 49
.............................................................................. 53
.................................... 58
.................................................... 66
....................................... 69
........................... 71
.................................................................................................. 78
.......... 79
................................................................................................................. 84
coscoscoscos
yyyp
112
=+++=
θθθθ
sinsinsinsin
yyyp
112
=+++=
θθθθ
coscoscoscos
yyyp
212
242
=+++=
θθθθ
sinsinsinsin
yyyp
212
242
=+++=
θθθθ
coscoscoscos
ynynypn
ykn
npk
=+++=
θθθθ
sinsinsinsin
ynynypn
ykn
npk
=+++=
θθθθ
cos
sinsin
mmmm
πππ
+=+

rabarbr
mmmmmmmmm
=+==
sin;
cos
Aab
===
∑∑∑
coscos
sincos
πππππ
111


=⋅=⋅=
coscos
,,,,
aqn
m
n
q
1
ππππ
()sin
cos
sinsin
=++


===
∑∑∑
cossin
sinsin
πππππ
111
=⋅=⋅=
sinsin
,,,,
bqn
m
n
q
1
yAp
⋅=⋅=
⋅=⋅=
cos
,,,,,
,,,,
=⋅=
=⋅=
cos,,,,,
sin,,,,.
(10)
=⋅=⋅⋅
p
yyyy
=++++=
123
(5)
coscoscoscos
k
k
ππππ
⋅==+
∑∑∑
cos
sinsinsincos
k
k
ππππ
⋅==−
∑∑∑
cossinsin
mqx
qmx
k
k
ππππ
∑∑∑
111
cossin


:
⋅=⋅
⋅=⋅
(),
cos()cos,
sin()sin.
(6)
cos
xAa
=++

=⋅++
Apa
(7)
ππππ
()cos
cos
sincos
=++
yxxAa
−==++
−⋅==
−⋅==
(),()
sin;
()cos,,,,;
()sin,,,,.
012
012
(3)
(3),
cos

sin


cossin
k
i
i
m
p
k
∑∑∑∑
+===
111
=+++=⋅
eeee
,,,


m
m
p
2

cossin
k
+=⇒
(4)
⇒==
cos;sin
k
coscos;sinsin;cossin
ππππππ
∑∑∑
111
,,,
,,,
coscoscos
cos
mqx
mqx
k
k
ππππ
∑∑∑
111
sinsincos
mqx
mqx
k
k
ππππ
∑∑∑
111
(1)


kxpl
012
====⋅==⋅===
;;;;;;
αααα
yyyyyy
0120
;;;;;;()


πππ
cos
xAa
x
=+++++
++++
(2)
Aaaabbb
1212



(2)
pkk
1

yyyy
123
,,,,
123
,,,,
(2),
Aaaabbb
1212
aaxaxax
012
+++++
(5)
bbxbxbx
012
+++++
(5)
(6)
cxccxcxcx
=+++++
012
cxaxbxx
∑∑∑
000
,(,)
(8)






fxdxa
dxn
();()cos;
()sin,,,.
(1)







6

ccxcxcxcx
012
++++++
(1)
(,)
(1)

(1)

fxfR
()()
ccRcRcR
012
+++++

ccRrcRrcRrfR
012
+⋅+⋅++⋅+→
fRrfR
()()
⋅→
(,)


fxfR
()()
aaaa
012
(2)
bbbb
012
(3)
cababab
nnnn
0110
cccc
012
(4)
:

=±π
Sxr
()()
−++−
[,]
()()
(,)()
[,]

()()
[,)
()(),(,)
∈−∞+∞
gtR




. 1, 4).

(,)




Sxr
gxgx
Sxr
fxfx
()()
()()
++−
++−
→−→−
1010
(,)




SxrgxSxrfx
(,)()(,)()
→
→−→−
1010
fgfg
()(),()()
−+=
0000
=±π
Sxr
Sxr
()()
(,)
()()
−++−
−++−
→−→−
1010
ππππ
[,]

()()
(,)()
∈−∞+∞
(,)
(,)()
[,]
(13)
2
2
rrt
rrt
()sin()sin
(9)
2
2
rrt
()sin
rrt
()sin
sinsin

rrt
()sinsin
(13)
Sxr
fxfx
()()
sin
++−
2
0
2
0
2
2
2

Sxr
fxfx
()()
++−
1),
3)
0



[,]
ftR
()[,]
∈−ππ
−


Sxr
fxfx
()()
++−


Sxrfx
(,)()
10000
+++
1
rrt
()sin
ftR

(9)
fxfx
()()
(8)
Sxr
fxfx
()()

[][]
=+−++−−−
2020
fxtfxfxtfx
rrt
()()()()
()sin

0
fxtfxfxtfx
()()()()
2020
(10)
∫∫∫
fxtfxfxtfx
()()()()
2020
fxtfxfxtfxM
()()()()
+−+
20204
Mft
sup()





:
Sxr
fxfx
()()
++−
rrt
rrt
()sin()sin
Sxrft
rtxr
(,)()
cos()
−−+
Sxrfxu
rur
(,)()
cos
(

Sxrfxt
rtr
(,)()
122
Sxrfxt
rtr
(,)()
cos
122
122
fxt
rtr
Sxrfxt
rtr
(,)()
cos
122
122
fxt
rtr
cos
Sxrfxtfxt
rtr
(,)()()
cos
=++−
122
(7)
cossin
212
122121214
22222
−+=−−+=−+
rtrrtrrrt
cos(sin)()sin
Sxrfxtfxt
rrt
(,)()()
()sin
=++−
ftR
(8).
anxbnxM
cossin
(4)
AMr
SxrAranxbnx
(,)(cossin)
=++
Aab
Aftdtaftntdtbftntdt
===
−−−
∫∫∫
πππ
();()cos;()sin
Sxrftdtftrntxdt
(,)()()cos()
=+−=
=⋅+−
ftrntxdt
()cos()
+−=⋅
−−+
rntx
rtxr
cos()
cos()
Sxr
rtxr
ftdt
cos()
−−+
ftR
:


Sxr
fxfx
()()


Sxrfx
(,)()
(,)()
++++
rrr
coscoscos
ααα
Srrr
=++++
coscoscos
ααα
2222
223
rSrrr
coscoscoscoscos
ααααα
⋅=+++
2coscoscos()cos()
=−+
2123
rSrrr
coscos(cos)(coscos)
ααααα
⋅=+++++⇒
⇒⋅=+++++++
21223
2223
rSrrrrrr
cos(coscos)(coscoscos)
αααααα
(coscos)
+++=+
rrS
(coscoscos)
rrrS
ααα
+++=−
rSrSS
cos
α⋅=+
112
()(cos)
cos
−=⋅−+⇒=⋅
rSrrS
ftR
Aanxbnx
(cossin)
Aranxbnx
(cossin)
(4)
ftR

0
ftM
aftntdtMbftntdtM
=≤=≤
()cos;()sin

0
SrSrSSrrr
()()()
−−−+−

SSrSS
−−
n
1
SrSrSS
()()
−−−+
2
SSA

SrSrA
()()()
−−+
1,
()()

SrS
SrS
2345
−+−+−+
rrrrr
(6)
1
1
lim()lim
→−→−
1010
(5)


++++=⋅
rrr
coscoscos
ααα
aSS
110
aSS
221
aSS
nnn
KKKKKK
KKKKKK
SrSSSrSSrSSr
()()()()
=+−+−++−+
01021
SrSSrSrSrSrSrSr
()()()()()
=−+−+−++−+
0102
+++++
SrSrSr
(3)
SSrSrSr
012
+++++
(3)
(4)
(1)
n

012
,,,
MrMrMr
++++
MMrMrMr
+++++
4
(3),
(4)
(3),
SrSrrSrSrrSr
()()()
=−⇒=−
∑∑∑
000
rrSrSr
∑∑∑
⇒=−⇒=−⋅
000
111
()()
SrSrSSrSrSrSSr
()()()()()
−=−−⇒−−−

1




[,]
anxbnxab
nnnn
cossin
+≤+
Aanxbnx
(cossin)
[,]
−ππ
5
[,]
−ππ
[,]
−ππ
aaaa
012
aararar
012
+++++
(2)
(1)
(2);
(2)
1

SSr
lim()
(1)

(1)
n
(

0

012
,,,
(2)
KKrKrKr
+⋅+⋅++⋅+
(2)

Saaaa
012





[,]


()()
[,]

[,]
−ππ
[,]
6.
[,]

()()
[,]
−ππ


afxnxdxafxd
=⇒=
()cos()
fxdx
sinsin

bfxnxdxbfxd
=⇒=−
()sin()
cos
fxdx
coscos

n
n
n
n
nnnn
≤+≤+
fxC
()[,]
fxR
()[,]

Aab

Ppq


()()
222
fxgxdxAPapbq
kkkk
()()()()()
−=−+−+−
fxgxdx
()()
fxgx
()()~

fxgx
()()~

-




()()
[,]
122
,cos,sin,cos,sin,
(1)

Axdxaxkxdxk
=====
012
();()cos(,,)
bxkxdxk
===
012
()sin(,,)
2:


0
[,]
4,
(1)
5.
[,]

()()
[,]

[,]

[,]
−ππ

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