**Delta - funksiya** yoki **Dirak delta funksiyasi**

Delta-funksiyaning sxematik grafigi

### Fanga kiritilish tarixi va matematik isboti

Birinchi marta 1926-yilda Dirak tomonidan kiritilgan {\displaystyle \delta }-funksiya odatda quyidagicha ta'riflanadi:

1. {\displaystyle \delta (x)=0}, agar {\displaystyle x\neq 0;(1)}
2. {\displaystyle \delta (x)=\infty }, agar {\displaystyle x=0;(2)}
3. {\displaystyle \int \limits \_{-\infty }^{+\infty }\delta (x)dx=1}, agar {\displaystyle -\infty <x<+\infty ;(3)}

Integrallash chegaralari {\displaystyle -\infty ,+\infty } bo`lishi shart emas. Delta-funksiya cheksiz bo`lgan nuqtani o`z ichiga olgan har qanday soha integrallash sohasi bo`lishi mumkin. Delta-funksiyaning ma'nosi ham shundaki, integral uning argumenti bo`yicha olinadi. 

Har qanday uzluksiz {\displaystyle f(x)} funksiya uchun yozish mumkin:

{\displaystyle \int \limits \_{-\infty }^{+\infty }f(x)\delta (x)dx=f(0)}, agar {\displaystyle -\infty <x<+\infty ;(4)}

Haqiqatdan, {\displaystyle \delta (x)} xususiyatiga muvofiq, yuqoridagi integralda faqat {\displaystyle x=0} nuqta atrofigina ahamiyatlidir. Cheksiz kichik sohada uzluksiz funksiyani o`zgarmas hisoblash mumkin. U vaqtda {\displaystyle f(x)} funksiyaning {\displaystyle f(0)} qiymatini integral belgisining oldiga chiqariladi. Qolgan integral esa 3-formulaga asosan birga tengdir.

Delta-funksiya uchun muhim bo`lgan yana bir formulani qarab ko`raylik:

{\displaystyle \delta (kx)={\frac {1}{|k|}}\delta (x);(5)},
agar {\displaystyle k=const\neq 0}

Haqiqatdan ham, agar {\displaystyle kx} ni {\displaystyle \xi } orqali belgilasak,

{\displaystyle \xi =kx;(6)}

  

U vaqtda (1) va (2) formulalarga asosan,

{\displaystyle \delta (\xi )=0,} agar{\displaystyle \xi \neq 0;(7)}
{\displaystyle \delta (\xi )=\infty ,} agar {\displaystyle \xi =0;(8)}

  

(5) da ifodalangan tenglik belgisining ma'nosi shundan iboratki, uning o`ng va chap tomonlari integral ostidagi ko`paytuvchilar sifatida olinib, bir xil natija beradi. 

Tenglamaning chap tomonini ko`rib chiqaylik. Agar {\displaystyle k>0} bo`lsa,

{\displaystyle \int \limits \_{-\infty }^{+\infty }\delta (kx)dx={\frac {1}{|k|}}\int \limits \_{-\infty }^{+\infty }|k|\delta (kx)dx={\frac {1}{|k|}}\int \limits \_{-\infty }^{+\infty }k\delta (kx)dx={\frac {1}{|k|}}\int \limits \_{-\infty }^{+\infty }\delta (kx)d(kx)} 
bo`ladi. Ammo (6) ga muvofiq,

{\displaystyle \int \limits \_{-\infty }^{+\infty }\delta (kx)d(kx)={\frac {1}{|k|}}\int \limits \_{-\infty }^{+\infty }\delta (\xi )d\xi }

Endi (6)-(8) ifodalarni nazarda tutib, delta-funksiya ta'rifiga asosan yozishimiz mumkin

{\displaystyle \int \limits \_{-\infty }^{+\infty }\delta (\xi )d\xi =1;(9)}

demak,

{\displaystyle \int \limits \_{-\infty }^{+\infty }\delta (kx)dx={\frac {1}{|k|}}}

Agar {\displaystyle k<0} bo`lsa, 

{\displaystyle \int \limits \_{-\infty }^{+\infty }\delta (kx)dx={\frac {1}{|k|}}\int \limits \_{-\infty }^{+\infty }|k|\delta (kx)dx=-{\frac {1}{|k|}}\int \limits \_{-\infty }^{+\infty }k\delta (kx)dx=-{\frac {1}{|k|}}\int \limits \_{-\infty }^{+\infty }\delta (kx)d(kx)=-{\frac {1}{|k|}}\int \limits \_{-\infty }^{+\infty }\delta (\xi )d\xi ={\frac {1}{|k|}}\int \limits \_{-\infty }^{+\infty }\delta (\xi )d\xi ={\frac {1}{|k|}}} 
bo`ladi.

(5) ning o`ng tomonidan olingan integral esa

{\displaystyle \int \limits \_{-\infty }^{+\infty }{\frac {1}{|k|}}\delta (x)dx={\frac {1}{|k|}}\int \limits \_{-\infty }^{+\infty }\delta (x)dx={\frac {1}{|k|}}} 
bo`ladi.

Shunday qilib, (5) ifoda isbotlandi. Xususiy hol {\displaystyle k=-1} uchun 

{\displaystyle \delta (-x)=\delta (x);(10)}

### Xossalari

* Delta-funksiya - juft funksiya hisoblanadi.
* {\displaystyle x\delta ^{\prime }(x)=-\delta (x)}
* {\displaystyle \delta (f(x))=\sum \_{k}{\frac {\delta (x-x\_{k})}{|f'(x\_{k})|}}}, bu yerda {\displaystyle x\_{k}} - {\displaystyle f(x)} funksiyaning nollari
* Bir o`lchamli Delta-funksiyadan olingan integral Heaviside funksiyasi beradi:

{\displaystyle {\displaystyle \theta (x)=\left\{{\begin{array}{\*{35}{l}}0,&x<0,\\1,&x>0.\\\end{array}}\right.}}

* Delta-funksiyaning filtrlash xossasi:

{\displaystyle \int \limits \_{-\infty }^{+\infty }f(x)\delta (x-x\_{0})\,dx=f(x\_{0}).}

### Integral tasavvurlar

Amaliyotda, ko`pincha, Delta-funksiyaning integral ko`rinishidan foydalanish qulay:

{\displaystyle :\delta (t)={\frac {1}{2\pi }}\int \limits \_{-\infty }^{+\infty }e^{i\omega t}\,d\omega }

Isbot  

Quyidagi integralni ko`rib chiqaylik:

{\displaystyle I(t)={\frac {1}{2\pi }}\int \limits \_{-\infty }^{\infty }e^{i\omega t}\,d\omega ,}    (1)
Ushbu integralni quyidagi limit ko`rinishida ham yozishimiz mumkin:

{\displaystyle I(t)=\lim \_{N\to \infty }I\_{N}(t),}
bu yerda

{\displaystyle I\_{N}(t)={\frac {1}{2\pi }}\int \limits \_{-N}^{N}e^{i\omega t}\,d\omega ={\frac {1}{\pi }}N{\frac {\sin {tN}}{tN}}.}    (2)
Ma'lumki,

{\displaystyle \int \limits \_{-\infty }^{\infty }{\frac {\sin t}{t}}\,dt=\pi .}    (3)
Agar ixtiyoriy {\displaystyle N} uchun (3) dan foydalansak, quyidagi tenglik o`rinli:

{\displaystyle \int \limits \_{-\infty }^{\infty }I\_{N}(t)\,dt={\frac {1}{\pi }}\int \limits \_{-\infty }^{\infty }{\frac {\sin {tN}}{tN}}\,d(tN)=1.}    (4)
Demak, *N* ning cheksiz o`suvchi qiymatlarida (2) funksiya uchun Delta-funksiyaning barcha xossalari o`rinli va u {\displaystyle \delta (t)} ga intiladi.

Yana qarang
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* Green-funksiyasi
* Bessel-funksiyasi

Adabiyotlar
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* *Mallin R.X.*, Klassik elektrodinamika, O`qituvchi, - T. 1974
* *Дирак П. А. М.* Основы квантовой механики / Пер. с англ. — М. 1932 (есть много переизданий).