Heaviside step function and Dirac delta function: Foundations

4.6.31 · D1 · Maths › Ordinary Differential Equations › Heaviside step function and Dirac delta function

Ye page assume karta hai ki tumhe kuch nahi pata. Parent note ko aaram se padhne se pehle, usme use hone wali har squiggle pehle ek picture ban jaani chahiye. Hum unhe ek ek karke build karte hain, har ek pichli par tikaa hua.

0. "Function of time" hota kya hai

Hume iski zaroorat kyun hai. Parent page par har cheez time ki function hai: ek switch state, ek push, ek spring ki hilti hui position. Agar "graph = time across, value up" tumhare liye automatic nahi hai, toh baad ki koi bhi cheez samajh nahi aayegi. Neeche har figure mein axes dekho — time hamesha left se right chalti hai.

Figure — Heaviside step function and Dirac delta function

1. "Pehle/baad" ki idea: piecewise

Topic ko iski zaroorat kyun hai. Real inputs khaas moments par apna behaviour change karte hain — ek switch pehle kuch nahi karta, phir kuch karta hai. Aage hum khaas "before/after" functions (ek switch, ek kick) se milenge, lekin general idea bas yahi hai: time ko ek marker par split karo aur har side par alag rule use karo.

Yahan sabse simple possible picture hai — do flat segments jo marker $a$ par jump karte hain:

2. Laplace variable $s$ — hum ise kyun invent karte hain

Kisi bhi integral se pehle, hume letter $s$ kamana hoga, kyunki yeh neeche ki har cheez ke andar aata hai.

Intuition

$s$ kya hai aur kyun exist karta hai Hum ek time-function ko ek fading multiplier $e^{- s t}$ se (§3 mein build hua) weight karne wale hain taaki saare future time par uska total area finite aaye. Woh knob jo control karta hai ki multiplier kitni tezi se fade ho, ek naya number hai jise hum $s$ kehte hain. $s$ ko ek decay-rate dial samjho: badhao ise, aur door ka future zyada squash hota hai. Parent page par har cheez ke liye hum $s$ ko ek real number $s > 0$ rakhte hain (positive, taaki weighting actually decay kare).

$s$ actually kahan rehta hai? Zyada full theory mein $s$ ko ek complex number hone diya jaata hai, aur $s$ ka woh set jiske liye weighted area finite hai use region of convergence kehte hain — zyada general functions ke liye " $s > 0$ " ka honest statement. Hum real $s > 0$ par hi rehte hain; complex plane aur convergence regions Laplace Transform — Definition and Existence handle karta hai.

Ab ki $s$ ka ek matlab hai, §4 mein integral saaf padha jayega.

3. Graph ke neeche ka area: integral $\int$

Picture: curve ke neeche ke region ko width $d t$ aur height $f (t)$ ki patli vertical strips mein kaato; har strip ka area $f (t) d t$ hai; $\int$ unhe sab jod deta hai.

Topic ko iski zaroorat kyun hai — do reasons.

Dirac delta (§7 mein aane wala) sirf uske area ke kaam se define hota hai — "total push = spike ke neeche ka area = $1$ ". Koi integral nahi, koi delta nahi.
Laplace transform (§4) ek integral hai.

Formula Woh ek integral jo tumhe karna aana chahiye — aur

kyun $\int e^{- s t} d t = \frac{e ^{- s t}}{- s} .$ Coefficient $- \frac{1}{s}$ kyun hai? Integrate karna differentiating ko undo karta hai, toh hum poochh te hain: "kaunsi function ka derivative $e^{- s t}$ hai?" $e^{- s t}$ ko chain rule se differentiate karne par front mein ek factor $- s$ aa jaata hai: $\frac{d}{d t} e^{- s t} = - s e^{- s t}$ . Yeh extra $- s$ unwanted hai, toh hum $- s$ se pre-divide karte hain ise cancel karne ke liye — phir $\frac{d}{d t} (\frac{e ^{- s t}}{- s}) = e^{- s t}$ , exactly jo chahiye tha. Coefficient $- \frac{1}{s}$ exactly woh mess hai jo chain rule karta hai, ise saaf karne ke liye daala gaya hai.

$a$ se $\infty$ tak evaluate karte hue ( $s > 0$ use karke, toh $e^{- s \cdot \infty} = 0$ ): $\int_{a}^{\infty} e^{- s t} d t = [\frac{e ^{- s t}}{- s}]_{a}^{\infty} = 0 - \frac{e ^{- a s}}{- s} = \frac{e ^{- a s}}{s} (s > 0) .$ Yeh akela computation parent page par $L {u (t - a)} = e^{- a s} / s$ produce karta hai.

4. Exponential $e^{- s t}$

Neeche ki picture teen decay curves dikhati hai: ek slide jo tezi se girta hai phir flat ho jaata hai, axis ko kabhi bilkul nahi chhoota, bade $s$ ke liye steeper.

Topic ko iski zaroorat kyun hai. $e^{- s t}$ Laplace transform ke andar "weighting" hai. Yeh functions ko tame karta hai taaki unka infinite-time area finite ho — exactly yahi reason hai ki §3 mein integral sirf $s > 0$ ke liye converge karta hai.

5. Laplace transform $L {\cdot}$ aur $F (s)$

Picture: do side-by-side worlds. Left par, time $t$ mein ek graph. $L$ labelled ek tunnel ise right par le jaata hai, jahan yeh $s$ mein ek graph ban jaata hai.

Topic ko iski zaroorat kyun hai. Differential equations time world mein mushkil hote hain. Laplace transform calculus (derivatives, delays) ko $s$ -world mein algebra (multiply, divide) mein badal deta hai. Tum easy algebra solve karte ho, phir tunnel se wapas aate ho. Laplace Transform — Definition and Existence dekhein jab integral exist karne ki permission ho, aur Solving IVPs with Laplace Transforms round-trip in action ke liye.

6. Delay stamp $e^{- a s} F (s)$ (time shift = multiply kyun)

Picture: wahi same wiggle, time axis par daayein slide ki gayi; $s$ -world mein sirf ek constant multiplier $e^{- a s}$ aata hai.

Mandatory gate $u (t - a)$ kyun? Kyunki "late bajao" ka matlab time $a$ se pehle silent rehna bhi hai. Gate hataao aur tumhara answer galat tarike se event se pehle baj uttha — parent page ki sabse zyada warn ki gayi galti. (Gate $u$ agle, §7 mein define hota hai.)

7. Derivative $\frac{d}{d t}$ aur "slope"

Picture: graph ko ek point par zoom karo; derivative woh tiny line ki tilt hai jo tumhe dikhti hai.

Topic ko iski zaroorat kyun hai. Yahi woh hinge hai jo ek switch ko ek kick mein badalta hai — lekin yeh dekhne ke liye pehle switch aur kick khud chahiye, jo hum ab define karte hain.

8. Steps aur impulses ke liye specific symbols

9. Unifying fact: $δ$ , $u$ ki slope hai

Ab ki $u$ , $δ$ , aur $\frac{d}{d t}$ teeno exist karte hain, hum unhe join kar sakte hain. Claim hai: $\frac{d}{d t} u (t - a) = δ (t - a) .$ Ise assert karne ki jagah hum ise ek finite "honest" version aur ek limit use karke dheere dheere build karte hain.

Step 1 — KYA. Ramp ka total climb $1$ hai, toh uske slope-box ka area $width \times height = ε \cdot \frac{1}{ε} = 1$ hai. $ε$ jo bhi ho, box ka area exactly $1$ rehta hai.

Step 2 — KYUN. Switch ko sharper karna matlab $ε$ ko shrink karna. Jab $ε \to 0^{+}$ ramp true instantaneous cliff ban jaata hai $u$ ka, aur uska slope-box taller ( $1/ ε \to \infty$ ) aur thinner (width $\to 0$ ) hota jaata hai jabki uska area $1$ par pinned rehta hai.

Step 3 — KAISA DIKHTA HAI. Woh limiting object — har jagah zero, ek instant par infinite, area exactly $1$ — exactly wahi amber spike $δ (t - a)$ hai §8 se. Toh step ki slope hi delta hai. Neeche ki figure $ε$ ko wide se narrow tak walk karati hai taaki tum box ko spike mein grow hote dekh sako.

10. Ye tools topic ke liye chain mein kaise jud te hain

Map se pehle do aur naam file karne hain: Piecewise and Periodic Forcing Functions woh hai jo §1 + §8 build karte hain (steps se messy inputs likhna), aur Impulse Response and Convolution woh hai jo §8 ka $δ$ unlock karta hai (ek akele kick par system ka jawab).

Map ko top-to-bottom padho jaise "kya kya feed karta hai": raw ideas (function, integral, exponential, derivative) top ke paas hain; har arrow matlab hai "head se pehle tail chahiye"; sab kuch neeche steps aur impulses ke saath IVPs solve karne mein funnel hota hai, jo parent topic ka goal hai.

Equipment checklist

Answers chhupao; tum ready ho sirf tab jab har reveal wahi ho jo tumne kaha.

Kya main ek graph ko "time across, value up" padh sakta hoon?

Haan — horizontal axis time

t

hai, vertical axis output

f (t)

hai.

Jump par, boundary

t = a

par bilkul function ki value kya hai?

"Don't care" — half-open convention se fix karo; ek akela point zero area add karta hai, toh integrals aur Laplace transforms unaffected rehte hain.

Letter

s

kya hai, aur woh kahan rehta hai?

Ek decay-rate dial; yahan real

s > 0

rakha gaya, lekin generally complex, convergence ke region ke saath.

\int_{p}^{q} f (t) d t

ka ek picture mein kya matlab hai?

f

ke graph aur time-axis ke beech ka signed area,

t = p

t = q

tak.

\int e^{- s t} d t

mein coefficient

- 1/ s

kyun hai?

Chain rule

\frac{d}{d t} e^{- s t} = - s e^{- s t}

banata hai;

- s

se pre-divide karna us unwanted factor ko cancel karta hai.

s > 0

ke liye

\int_{a}^{\infty} e^{- s t} d t

kya hai?

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

e^{- s t}

(

s > 0

ke saath)

t

badhne par kya karta hai?

1

se start hota hai aur smoothly

0

ki taraf decay karta hai; bada

s

tezi se decay karta hai.

L {f} = F (s)

calculus ko kya mein badal deta hai?

s

-world mein algebra — derivatives aur delays multiply/divide ban jaate hain.

L^{- 1}

kya karta hai?

Return tunnel:

s

ki function se time-function recover karta hai.

a

ka time-delay

e^{- a s}

factor kyun deta hai?

Signal ko

a

seconds baad slide karna uske transform ko

e^{- a s}

se multiply karne ke roop mein dikhta hai.

e^{- a s} F (s)

invert karte waqt gate

u (t - a)

kyun appear karna chahiye?

Yeh delayed signal ko

t = a

se pehle silent rakhta hai; iske bina answer

t < a

ke liye galat hai.

\frac{d}{d t}

kya measure karta hai?

Slope — function har instant par kitni tezi se change ho rahi hai.

δ (t - a) = \frac{d}{d t} u (t - a)

kyun hai?

Step ka ek hi change ek instantaneous jump of

1

hai; uska slope-box har

ε

ke liye area

1

rakhta hai aur

ε \to 0

hone par infinite-thin spike ban jaata hai.

Sifting property batao.

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

δ (t - a)

precisely kya hai?

Pointwise-valued function nahi — ek distribution/rule jo sirf uske integral (sifting) se define hota hai, area

1