Heaviside step function and Dirac delta function

4.6.31 · Maths › Ordinary Differential Equations

1. The Heaviside Step Function

WHY it matters. Yeh hume piecewise inputs ko ek formula mein likhne deta hai. Chahte ho ek function jo $g (t)$ ho lekin sirf time $a$ ke baad? Multiply karo: $g (t) u (t - a)$ , yeh $t < a$ ke liye use khatam karta hai aur $t > a$ ke liye reveal karta hai.

HOW to build windows. Kisi cheez ko $a$ par on aur $b$ par off karne ke liye: $u (t - a) - u (t - b) = {10 a < t < b otherwise$ Yeh "boxcar" piecewise modelling ka kaam ka haath hai.

Laplace transform of the step

2. The Dirac Delta Function

HOW it arises as a limit. Ek narrow box define karo width $ε$ , height $1/ ε$ (area $= 1$ ), $a$ ke centre par: $δ_{ε} (t - a) = \frac{1}{ε} [u (t - a) - u (t - a - ε)] .$ Jaise $ε \to 0^{+}$ , box infinitely tall aur thin ho jaata hai, area abhi bhi $1$ — wahi $δ (t - a)$ hai.

Sifting property — box se derive kiya

Laplace transform of the delta

Figure — Heaviside step function and Dirac delta function

3. Common Mistakes (Steel-manned)

4. Active Recall

Recall Quick self-test (hide and answer)

Sifting property state karo aur $L {δ (t - a)}$ derive karo.
$δ = \frac{d}{d t} u$ kyun hai?
$(t - 3)^{3} u (t - 3)$ transform karo.
Agar tum $L^{- 1} {e^{- a s} F (s)} = f (t - a)$ likhte ho toh kya missing hai?

Recall Feynman: 12-saal ke bachche ko explain karo

Step function ek light switch jaisi hai: bilkul andhera, phir click, poori tarah ujala — woh "click" ek exact moment par hota hai. Delta function woh click khud hai: ek super-quick poke jo ek fixed amount "push" deta hai bilkul bhi time nahi lete, jaise marble flick karna. Neat trick yeh hai: click par light ki brightness kitni tezi se badlti hai woh exactly woh poke hai. Toh poke (delta) sirf switch flipping ki speed hai (step ka slope). Maths mein, jab tum poochho "poke ne total kitna push diya?", jawab hamesha wahi hota hai jo system ne us instant mein feel kiya — delta us ek value ko pick out karta hai aur baki sab ignore karta hai.

u (t - a)

ki definition

t < a

ke liye

0

t > a

ke liye

1

;

t = a

par value

1

switch on karta hai.

L {u (t - a)}

\frac{e ^{- a s}}{s}

s > 0

ke liye.

Second shifting theorem

L {f (t - a) u (t - a)} = e^{- a s} F (s)

Delta ki sifting property

\int_{- \infty}^{\infty} f (t) δ (t - a) d t = f (a)

L {δ (t - a)}

e^{- a s}

(aur

L {δ (t)} = 1

Step aur delta ka relation

δ (t - a) = \frac{d}{d t} u (t - a)

(a, b)

par boxcar (window)

u (t - a) - u (t - b)

L^{- 1} {e^{- a s} F (s)}

f (t - a) u (t - a)

(gate zaroori hai!).

L {t u (t - 2)}

t = (t - 2) + 2

likho →

e^{- 2 s} (\frac{1}{s ^{2}} + \frac{2}{s})

δ

ek sach functional function kyun nahi hai

Yeh ek distribution hai jo sirf integrals ke zariye define hoti hai;

a

par iska koi finite pointwise value nahi hai.