Delta-funksiya

Delta - funksiya yoki Dirak delta funksiyasi

Delta-funksiyaning sxematik grafigi

Fanga kiritilish tarixi va matematik isboti

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

1. $\delta (x)=0$, agar $x\ne 0;(1)$
2. $\delta (x)=\mathrm{\infty }$, agar $x=0;(2)$
3. $\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (x)dx=1$, agar $-\mathrm{\infty }<x<+\mathrm{\infty };(3)$

Integrallash chegaralari $-\mathrm{\infty },+\mathrm{\infty }$ 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 $f(x)$ funksiya uchun yozish mumkin:

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}f(x)\delta (x)dx=f(0)$, agar $-\mathrm{\infty }<x<+\mathrm{\infty };(4)$

Haqiqatdan, $\delta (x)$ xususiyatiga muvofiq, yuqoridagi integralda faqat $x=0$ nuqta atrofigina ahamiyatlidir. Cheksiz kichik sohada uzluksiz funksiyani o`zgarmas hisoblash mumkin. U vaqtda $f(x)$ funksiyaning $f(0)$ 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:

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

Haqiqatdan ham, agar $kx$ ni $\xi$ orqali belgilasak,

$\xi =kx;(6)$

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

$\delta (\xi )=0,$ agar$\xi \ne 0;(7)$ $\delta (\xi )=\mathrm{\infty },$ agar $\xi =0;(8)$

(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 $k>0$ bo`lsa,

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (kx)dx=\frac{1}{|k|}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}|k|\delta (kx)dx=\frac{1}{|k|}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}k\delta (kx)dx=\frac{1}{|k|}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (kx)d(kx)$ bo`ladi. Ammo (6) ga muvofiq,

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (kx)d(kx)=\frac{1}{|k|}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (\xi )d\xi$

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

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (\xi )d\xi =1;(9)$

demak,

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (kx)dx=\frac{1}{|k|}$

Agar $k<0$ bo`lsa,

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (kx)dx=\frac{1}{|k|}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}|k|\delta (kx)dx=-\frac{1}{|k|}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}k\delta (kx)dx=-\frac{1}{|k|}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (kx)d(kx)=-\frac{1}{|k|}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (\xi )d\xi =\frac{1}{|k|}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (\xi )d\xi =\frac{1}{|k|}$ bo`ladi.

(5) ning o`ng tomonidan olingan integral esa

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\frac{1}{|k|}\delta (x)dx=\frac{1}{|k|}\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}\delta (x)dx=\frac{1}{|k|}$ bo`ladi.

Shunday qilib, (5) ifoda isbotlandi. Xususiy hol $k=-1$ uchun

$\delta (-x)=\delta (x);(10)$

Xossalari

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

$\theta (x)=\{\begin{array}{cc}0,& x<0,\\ 1,& x>0.\end{array}$

• Delta-funksiyaning filtrlash xossasi:

$\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}f(x)\delta (x-x_0)dx=f(x_0).$

Integral tasavvurlar

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

$:\delta (t)=\frac{1}{2\pi }\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}{e}^{i\omega t}d\omega$

Yana qarang

• Green-funksiyasi
• Bessel-funksiyasi

Adabiyotlar

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