Laplace transform — definition, region of convergence: Foundations

4.6.25 · D1 · Maths › Ordinary Differential Equations › Laplace transform — definition, region of convergence

Pehle aapko parent note (Laplace transform — definition, ROC) bina kisi confusion ke padhna hai, toh us mein aane wale har symbol par aapki pakad honi chahiye. Hum unhe ek-ek karke banate hain, har ek pichle se.

0. Characters, appearance ke order mein

Yeh poora cast hai jo parent note use karta hai. Hum har ek se neeche milenge.

Symbol	Padha jaata hai	Section
$t$	time	§1
$f (t)$	time ki ek function	§2
$\int_{a}^{b} \dots d t$	" $t$ pe add karo"	§3
$\int_{0}^{\infty}$	improper integral	§4
$e^{x}$ , $e^{- s t}$	exponential / kernel	§5
$s = σ + iω$	complex frequency knob	§6
$Re (s) = σ$	real part	§6
$L {f} = F (s)$	transform khud	§7
"converges", ROC, abscissa $a$	jab sum settle ho	§8
exponential order, $\leq M e^{a t}$	growth speed limit	§9

1. $t$ — time, input axis

Picture: ek horizontal number line jo right ki taraf point kar rahi hai. $t = 0$ "abhi" hai; har cheez right mein baad mein hai.

Topic ko iske kyon zaroori hai: jo signals hum transform karte hain (ek voltage, ek temperature, ek swinging pendulum) woh cheezein hain jo time ke saath change hoti hain. $t$ woh axis hai jis par woh rehti hain.

2. $f (t)$ — time ki ek function

Picture: time line ke upar ek curve. Floor par har $t$ ke liye, curve ki height hi $f (t)$ hai. Figure dekho: teen signals — ek flat ( $f = 1$ ), ek rising ( $f = t$ ), aur ek wiggling ( $sin$ ).

Figure — Laplace transform — definition, region of convergence

Topic ko iske kyon zaroori hai: $f (t)$ raw material hai — woh cheez jo transform machine mein jaati hai. Parent ke har example ( $1$ , $t$ , $e^{a t}$ , $sin ω t$ ) mein $f (t)$ ka ek specific choice hai.

3. $\int_{a}^{b} g (t) d t$ — "area add karo"

Picture: curve $g (t)$ jiske neeche ka region shaded hai aur rectangles mein kaata hua hai.

$\int$ ek stretched "S" hai Sum ke liye.
$d t$ ek slice ki tiny width hai.
$a$ (bottom) aur $b$ (top) limits hain — adding kahan shuru aur kahan khatam hoti hai.

Topic ko iske kyon zaroori hai: Laplace transform ek integral hi hai. Yeh poore signal ko har $s$ ke liye ek single number mein add karta hai. Agar aap "area = sum of strips" picture nahi kar sakte, toh definition sirf squiggles hai.

4. $\int_{0}^{\infty}$ — improper integral

Picture: shaded area hamesha ke liye rightward extend hota rehta hai. Do cheezein ho sakti hain:

running total ek fixed height ki taraf settle karta hai (hum kehte hain yeh converges), ya
yeh infinity ki taraf bhaag jaata hai (yeh diverges).

Topic ko iske kyon zaroori hai: Laplace integral $\infty$ tak jaata hai. Yeh converge karega ya nahi — yahi Region of Convergence ka poora drama hai. Full machinery ke liye Improper Integrals dekho.

5. $e^{x}$ aur kernel $e^{- s t}$

Picture: do curves — ek floor ki taraf slide kar rahi hai (decay), ek upar rocket kar rahi hai (growth). Figure dekho.

Topic ko iske kyon zaroori hai: $e^{- s t}$ transform ka dil hai — "fade-out filter." ROC ke baare mein sab kuch $e^{- s t}$ ke fade hone aur $f (t)$ ke badhne ke beech ek race se aata hai.

6. $s = σ + iω$ aur iska real part $Re (s)$

Picture: ek 2D plane (" $s$ -plane"). Horizontal axis $= σ$ , vertical axis $= ω$ . $s$ ki har choice is plane par ek point hai.

Woh ek fact jo convergence govern karta hai: $e^{- s t}$ ki size sirf $σ$ par depend karti hai, $ω$ par nahi: $∣ e^{- s t} ∣ = e^{- (σ + iω) t} = e^{- σ t} .$ $iω$ part sirf spin (rotate) karta hai, magnitude kabhi nahi badlata. Toh decay sirf $σ = Re (s)$ se control hota hai.

Topic ko iske kyon zaroori hai: ROC $s$ -plane mein ek region hai — specifically ek half-plane jo vertical line $σ = a$ se cut hoti hai. Aap us region ko $s$ -plane ke bina picture nahi kar sakte.

7. $L {f} = F (s)$ — transform

Picture: ek box labeled $L$ . Andar jaati hai time pe curve $f (t)$ ; bahar aati hai $s$ -plane pe curve $F (s)$ . Same information, nai language.

Topic ko iske kyon zaroori hai: parent literally yahi definition build karta hai. Pehle sab kuch scaffolding tha taaki yeh line plain English mein padhe: " $f$ ko $e^{- s t}$ se fade karo, use time pe poora add karo, aur total ko fade-speed $s$ ki function $F (s)$ ke roop mein record karo." Wapas jaana hai toh Inverse Laplace Transform dekho.

8. Convergence, ROC, aur abscissa $a$

Picture: $s$ -plane mein $σ = a$ par ek vertical dividing line. Iske right taraf sab safe hai (converges); left taraf sab diverge karta hai.

Kyun: agar $f$ $e^{a t}$ ki tarah grow karta hai, toh integrand $e^{- σ t} e^{a t} = e^{(a - σ) t}$ hai. Yeh sirf tab decay karta hai jab $a - σ < 0$ , yaani $σ > a$ . Jitni tez $f$ grow kare, utna aur right $s$ ko slide karna padega.

9. Exponential order — growth speed limit

Picture: curve $∣ f (t) ∣$ eventually ek ceiling $M e^{a t}$ ke neeche trapped hai. Jab tak $f$ kisi exponential ke neeche rehti hai, uska Laplace transform exist karta hai.

Topic ko iske kyon zaroori hai: yahi transform ke exist hone ki sufficient condition hai. $e^{t^{2}}$ jaisa monster har $e^{a t}$ ko beat karta hai — woh har ceiling ke through poke karta hai — isliye uska koi Laplace transform hi nahi hota. Dekho Exponential Order and Growth Rates.

Prerequisite map

Ise bottom-out padhein: time aur functions hamare paas integrate karne ke liye kuch deti hain; exponential kernel banata hai; complex $s$ fade-speed deta hai aur (apne real part ke through) decay decide karta hai; saath milkar woh transform produce karte hain, aur growth-vs-decay race ROC carve out karti hai.

Equipment checklist

Right side cover karo; kya aap reveal karne se pehle answer de sakte ho?

t \geq 0

hume kya restrict karta hai?

sirf future tak — signals jo "abhi,"

t = 0

se start hote hain.

f (t)

ek phrase mein?

ek rule jo time ke har moment ke liye ek output value deta hai.

\int_{a}^{b} g d t

pictorially kya compute karta hai?

g

ke neeche area, width

d t

ki skinny strips ke sum ke roop mein.

\int_{0}^{\infty}

ko improper kyun kaha jaata hai?

top limit infinite hai; hum finite area ka

lim_{T \to \infty}

lete hain.

Improper integral ke do outcomes?

yeh converges (finite number tak settle hota hai) ya diverges (infinity tak bhaag jaata hai).

e^{x}

ki defining superpower?

iska derivative khud ke barabar hai,

\frac{d}{d x} e^{x} = e^{x}

Kya

e^{- k t}

(jab

k > 0

) grow karta hai ya decay?

decay — yeh

1

0

ki taraf slide karta hai.

e^{- s t}

ke kernel hone ke do reasons?

yeh growth tame karta hai (decay), aur

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

d / d t

\times s

mein badal deta hai.

s

ko real/imaginary parts mein likho.

s = σ + iω

, jahan

σ = Re (s)

s

ka kaunsa part convergence control karta hai, aur kyun?

σ = Re (s)

, kyunki

∣ e^{- s t} ∣ = e^{- σ t}

(woh

iω

sirf rotate karta hai).

L {f}

words mein kya hai?

f

e^{- s t}

se fade karo, use time pe poora add karo, total ko

F (s)

ke roop mein record karo.

Har ROC ki shape?

ek right half-plane

Re (s) > a

Abscissa of convergence

a

kya hai?

boundary value; yeh signal ki growth rate ke barabar hota hai.

f

exponential order

a

ki hai jab...?

∣ f (t) ∣ \leq M e^{a t}

saare

t \geq T

ke liye, kuch

M > 0

T \geq 0

e^{t^{2}}

fail kyun karta hai?

yeh har

e^{a t}

ko outgrow karta hai, isliye koi kernel ise tame nahi kar sakta — transform kabhi converge nahi karta.

Connections

Parent topic (Hinglish)
Improper Integrals — §4 aur ROC ke peeche convergence machinery.
Exponential Order and Growth Rates — §9 ka growth-speed idea.
Inverse Laplace Transform — §7 ki machine ko undo karna.
Laplace Transform of Derivatives — §5 ki $\times s$ property cash karna.
Solving ODEs with Laplace Transforms — payoff.
Fourier Transform — woh special case jab $Re (s) = 0$ .

#flashcard