Science and e ducation

Aytaylik,
^f  ^x
funksiya ^{ , } da berilgan toq funksiya bo‘lsin:

^f ^x ^{  f}  ^x _{. U holda}

^f  ^x^{cos nx} _toq,
^f  ^x^{sin nx} _juft ^{n  1, 2,3,...}
funksiya bo‘ladi.

(1) formulalardan foydalanib, topamiz:
^{an } _{ } ^f  ^x^{cos nxdx } _{  } ^f  ^x^{cos nxdx } _ ^f  ^x ^{cos nxdx}_ ^
  0 
₁ 


^ ^ _ ^f  ^x^{cos nxdx  f}  ^x^{cos nxdx}_^{  0}
0
^{n  0,1, 2,}

₁  _{1 } 0  _
^{bn } _{ } ^f  ^x^{sin nxdx } _{  } ^f  ^x^{sin nxdx } _ ^f  ^x^{sin nxdx}_ ^


_{2 }
 0 


^ _{ } ^f  ^x^{sin nxdx}_
^{n 1, 2,}

 0 

Demak, toq
^f  ^x
funksiyaning Furye koeffitsiyentlari

^{an  0,} ^{n  0,1, 2,}
₂ 

bo‘lib, Furye qatori

^{bn } _ ^f  ^x^{sin nxdx,} ^{n  1, 2,}


0

bo‘ladi.
^f  ^x
funksiyaning Furye koeffitsiyentlarini

₁  _{1 } 0  _

2. 
   misol. Ushbu
   

^f  ^x <sup> x2



f</sup>  ^x ^~ _^{bn sin nx}
n1
^{  x   } juft funksiyaning Furye qatori topilsin.

Avvalo berilgan funksiyaning Furye koeffitsiyentlarini topamiz:

a₀ 


2
_ x²dx 
² ² ,

 ₀ 3

a_n 


2
_ x² cos nxdx 
2 sin nx ^
_x 2 _
⁴ _ x sin nxdx 


 ₀  n ₀ n ₀
^{4  x cos nx}  ^{1  } n ⁴

^ n ^ n
^ _{n } ^{cos nxdx} _ ^ ^1
^ _n2 ^. ^{n  1, 2,}

 ^{0 0} 

Demak,
^f  ^x ^{ x2}
funksiyaning Furye qatori

bo‘ladi.
2


^f  ^x ^{ x2 ~}
3



^{ 4}_^1
n1
n ^{cos nx} _n2

3. 
   misol. Ushbu
   

^f  ^x ^{ x}
^{  x  } 
toq funksiyaning Furye qatori topilsin.

Berilgan funksiyaning Furye koeffitsiyentlarini hisoblaymiz:

2 ^ 2 ^
x cos nx ^ 1 ^{
 2}^1n1

b_n  _{ } x sin nxdx  _{ }  _n  _{n } cos nxdx _  _n
0 _ ⁰ 0 _{ .}

Demak,
^{f  x  x} funksiyaning Furye qatori

bo‘ladi.
Faraz qilaylik, Ma’lumki, ushbu
^f  ^x
funksiya ^{ p , p}
 ^{p  0}
segmentda uzluksiz bo‘lsin.

almashtirish ^{ p , p}
t  ^ x p
oraliqni ^{ , } ga o‘tkazadi, ya’ni ^x o‘zgaruvchi ^{ p , p} da

o‘zgarganda ^t o‘zgaruvchi ^{ , } da o‘zgaradi. Endi
^f  ^x ^{ f  p t   } ^t ^.
 _ 

deymiz. Unda ^{ t }
funksiya ^{ , }
 
oraliqda berilgan uzluksiz funksiya bo‘ladi.

Bu funksiyaning Furye koeffitsiyentlari

a  ¹
n _


b  ¹
n _

_ ^ ^t ^{cos ntdt,}


_ ^ ^t ^{sin ntdt}

^{n  0 ,1, 2 ,}


<sup>n  1, 2 ,




f</sup>  ^x ^~ _
n1

ni topib, Furye qatorini yozamiz:

2
^ ^t  ^{~ a0 } _ ^a
cos nt  b
^{sin nt} 

Modomiki,
n n
n1 _.

ekan, unda
t  ^ x p

_ ^  _x ^ _~ a_{0  }^{ }
cos n ^ x  b sin n ^{ }

x ,

a
 _p  ₂  n
p ⁿ p ^

 
bo‘lib, uning koeffitsiyentlari
^n1  

1 ^{p }  ^ 

a_n 
_{p }  _{ p} x _cos n _p xdx ,
^{n  0,1, 2}

 p ^{ }
1 ^{p }  ^ 

b_n 
_{p }  _{ p} x _sin n _p xdx .
^{n  1, 2}

 p
bo‘ladi. Natijada ^{ p , p}
quyidagicha
 
da berilgan


^f  ^x


funksiyaning Furye qatorini

a₀ ^{ } n x n x ^
f  ^x ^{~ } _ ^{an cos  bn sin} _

2. 
   p p
   

^n1  
bo‘lishini topamiz, bunda

a_n 
_ ^f  ^x^{cos dx}

• 
  
  p
  
  p
  p
  

^{n  0,1, 2}

1 ^p n

b_n 
_ ^f  ^x^sin

• 
  
  p
  p
  

xdx
p
^{n  1, 2}

4. 
   misol. Ushbu
   

^f  ^x ^{ ex}
^{1  x  1}
funksiyaning Furye qatori topilsin.

Yuqoridagi formulalardan foydalanib, koeffitsiyentilarini topamiz:
^f  ^x ^{ ex}
funksiyaning Furye

^{1 1} n sin n x  cos n x ¹

a₀  _ e^xdx  e  e^1,
1
a_n  _ e^x cos n xdx 
1
1 n² ²
e^x 
1


 ¹ e cos n  e^1
1 n² ²
cos n _  _1_n
e  e^1
1 n² ²
^{n  1, 2 ,}

1

1
b_n 
1
_ e^x cos n xdx 
1
sin n x  n cos n x _x

1 n² ^{2 e }

_ 1
1 n² ²
_en cos n  ne^1 cos n _ 

Demak,
^n ^1n
^ 1 n² ²
_ n ^{e  e1}

_e ¹  e_  _1_ ¹
1 n² ²
^{n  1, 2,}

funksiyaning Furye qatori

^f  ^x ^{ ex}
^{1  x  1}

e  e^{1
 } ^1n
^1n1 ^

bo‘ladi.
e^x ~
 _e  e^1 _
2 _{n1 }1 n² ²
cos n 
1 n² ^{2
n sin n x}
_

Aytaylik,
^f  ^x _funksiya ^{a, b}
da berilgan bo’lsin. ^{a, b}
segment ^ak
nuqtalar

yordamida bo‘laklarga ajratilgan.
(a₀  a,

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