Science and Education

Keywords: Fourier series, Fourier coefficients. Fourier series expansion of functions.

1. Furye qatori.
Faraz qilaylik,


^f  ^x


funksiya


^{R } ^{,  }

da berilgan bo‘lsin. Ma’lumki,

shunday ^{T  R \ 0}
son topilsaki, ^{ x  R} da
^f  ^{x  T}  ^{ f}  ^x

tenglik bajarilsa,
^f  ^x
davriy funksiya, ^{T  0} son esa uning davri deyiladi.

Agar ^{T  0} son
^f  ^x
funksiyaning davri bo‘lsa, u holda
_kT  ^{k  1,  2,}

sonlar ham shu funksiyaning davri bo‘ladi.

Agar
^f  ^x _va
^g  ^x
davriy funksiyalar bo‘lib, ^{T  0} ularning davri bo‘lsa,
^f  ^x

^f  ^x ^{ g}  ^x^,
f  ^x^{ g}  ^x^,
g  ^x
_g _ x _  0_

funksiyalar ham davriy bo‘lib, ularning davri ^T ga teng bo‘ladi.

y  sin x,
y  cos x
funksiyalar ^{T  2}
davrli funksiya bo‘lgan holda ushbu

^  ^x ^{ a cos x  b sin x} _{( a, b, }
o‘zgarmas, ^{  0} )
_{T } 2

ham,
funksiya ham davriy funksiya bo‘lib, uning davri
^ bo‘ladi. Haqiqatan

 ^ x  ^{2 }  a cos ^ ^ x  ^{2 }  b sin ^ ^ x  ^{2 } 
^ _ ^  ^ _ ^  ^ _ ^
^{ }  ^{ }  ^{ }

bo‘ladi.
^{ a cos}^{ x  2}  ^{ b sin} ^{ x  2}  ^{ a cos x  b sin  x  }  ^x

Bu ataladi.
^{  x  a cos x  b sin x} sodda davriy funksiya bo‘lib, u garmonika deb

Aytaylik,
^f  ^x
funksiya ^{ , } da uzluksiz bo‘lsin. Unda

^f  ^x^{cos nx ,
f}  ^x^{sin nx}
^{n  1, 2,3,}

funksiyalar ham ^{ , }
da uzluksiz bo‘lib, ular ^{ , }
da integrallanuvchi

bo‘ladi. Bu integrallarni quyidagicha belgilaymiz:

0 _


a  ^1
 

1. 
   
   n _
   
   
    ¹
   

^ 

2. 
   
   n _
   
   
    ¹
   

^ 
^f  ^x <sup>dx,


f</sup>  ^x^{cos nxdx, f}  ^x^{sin nxdx.}
^{n  1, 2,}


^{n  1, 2,}
(1)

Bu sonlardan foydalanib, ushbu
^{a0 } _ <sup>a cos nx  b


sin nx</sup>

2
n n
n1
(2)

qatorni ( uni trigonometrik qator deyiladi) hosil qilamiz.
(2) qator funksional qator bo‘lib, uning har bir hadi garmonikadan iborat.

Ta’rif. (2) funksional qator munosabatlar bilan aniqlangan
^f  ^x
funksiyaning Furye qatori deyiladi. (1)

a₀ , a₁ ,b₁ , a₂ ,b₂ ,
sonlar Furye koeffitsiyentlari deyiladi.
7.2. Funksiyalarni Furye qatoriga yoyish.

Demak, berilgan
^f  ^x
funksiyaning Furye koeffitsiyentlari shu funksiyaga

bog‘liq bo‘lib, (2) formulalar yordamida aniqlanadi, qator esa quyidagicha:

2
^f  ^x ^{~ a0 } _ ^{a cos nx  b sin nx}

belgilanadi.

n n
n1 _.

1. 
   misol. Ushbu
   

^f  ^x ^{ ex} ^{  x  
,  0}
funksiyaning Furye qatori topilsin.

(1) formulalardan foydalanib, berilgan funksiyaning Furye koeffitsiyentlarini hisoblaymiz:

1
a₀  _

_{ e}

^xdx 
¹ _e _{ e} _ ²

 
 
sh ,

1
a_n 

_ e^x cos nxdx 
1  cos nx  n sin nx

2 2


_e x _

^ 
   n _

^ ^1n ^{1  2 sh}
  ²  n²
^{n  1, 2 ,}

1
b_n  _

_ e^x sin nxdx 


1. 
    sin nx  n cos nx
   

  ²  n²

_e x _


^ ^1n1 ^{1  2n sh}

^{n  1, 2 ,}

  ²  n²
Demak,

funksiyaning Furye qatori

^f  ^x ^{ ex}

^f  ^x ^{ ex ~ a0 } _ ^{a cos nx  b
sin nx} ^

2
n n
n1
2sh ^ 1 ^ ^1n ^
 _ ^ _2 ^ _{ 2  n2} ^{ cos nx  n sin nx}^

bo‘ladi.
_ ^n1 _

Aytaylik, U holda
^f  ^x
funksiya ^{ , }
da berilgan juft funksiya bo‘lsin:
^f ^x ^{ f}  ^x_.

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

(1) formulalardan foydalanib, topamiz:
^f  ^x
funksiyaning Furye koeffitsiyentlarini

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


_{1 } 
 0 


^ _{ } ^f  ^x^{cos nxdx } _ ^f  ^x^{cos nxdx}_ ^
 0 0 
₂ 


_ ^f  ^x^{cos nxdx}
0
^{n  0,1, 2,}

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


_{1 }  
 0 


^ _{ }^_ ^f  ^x^{sin nxdx } _ ^f  ^x^{sin nxdx}_ ^{ 0}
^{n 1, 2,}

 0 0 

Demak, juft
^f  ^x
funksiyaning Furye koeffitsiyentlari
₂ 


^{an } _ ^f  ^x^{cos nxdx}
0
^{n  0,1, 2,}

bo‘lib, Furye qatori

bo‘ladi.