Transforms of standard functions — proofs: Foundations

4.6.26 · D1 · Maths › Ordinary Differential Equations › Transforms of standard functions — proofs

Yeh page kuch bhi assume nahi karta. Agar tumne pehle kabhi $\int$ , $e^{- s t}$ , $L$ , $s$ , $\infty$ , ya $Re (s)$ nahi dekha, toh yahan se shuru karo aur upar se neeche padhte jao. Har symbol earn kiya jaata hai pehle, phir agla ushe use karta hai.

0. Woh poora sentence jo hum decode kar rahe hain

Parent topic ki har cheez is ek line par tiki hai:

$L {f (t)} = F (s) = \int_{0}^{\infty} e^{- s t} f (t) d t .$

Ab hum ise ek ek symbol karke, left se right, ek ek picture ke saath tod denge. Ant tak tumhe us line ke har mark par ungli rakh ke yeh kehna aana chahiye ki uska matlab kya hai aur woh kaisa dikhta hai.

Figure — Transforms of standard functions — proofs

Figure dekho: blue curve raw signal $f (t)$ hai, coral curve fading weight $e^{- s t}$ hai, aur shaded mint region unka product hai — woh cheez jiska area hum measure karte hain. Yeh picture mind mein rakho; neeche har symbol iska ek piece hai.

1. $t$ — input clock

Picture: ek number line jo point $0$ se shuru hoti hai aur hamesha right ki taraf jaati hai — $0, 1, 2, \dots$ Is kahani mein koi "zero se pehle" nahi hai; machine sirf $t \geq 0$ ki parwah karti hai.

Topic ko yeh kyun chahiye: ek Laplace transform un signals ke liye bana hai jo ek starting moment par "switch on" hoti hain (ek voltage jo $t = 0$ par apply hoti hai, ek mass jo $t = 0$ par release hoti hai). $t \geq 0$ tak restrict karna hi integral ki lower limit ko saaf $0$ banata hai.

2. $f (t)$ — signal, ek machine jo jawab deti hai "time $t$ par value kya hai?"

Picture: time axis ke upar ek curve bani hui. Curve ki unchai point $t$ ke upar hi value $f (t)$ hai. Kuch examples jo tum dekhoge: flat line $f (t) = 1$ , wiggle $f (t) = sin (t)$ , runaway growth $f (t) = e^{t}$ .

Topic ko yeh kyun chahiye: $f (t)$ raw material hai — woh cheez jo transform ho rahi hai. Results ki poori dictionary ( $L {1}$ , $L {t^{n}}$ , $L {sin a t}$ …) sirf is ek hi machine ko alag alag curves dena hai.

3. $e$ aur exponential $e^{x}$ — constant-percentage change

Picture: do mirror-image curves.

$e^{x}$ (jab $x > 0$ ): ek curve jo tezi se tezi se upar curl karti hai — explosive growth.
$e^{- x}$ (jab $x > 0$ ): ek curve jo height $1$ se shuru hoti hai aur floor $0$ ki taraf slide karti hai, kabhi use touch nahi karti.

Topic ko yeh kyun chahiye: Laplace transform ka poora trick fading factor $e^{- s t}$ hai. Decaying weight ke bina, ek signal ko infinite time tak add karna usually infinity deta. Exponential hi usse tame karta hai.

4. $s$ — tuning knob (aur yeh complex kyun ho sakta hai)

Picture: coral weight-curves ka ek family socho, har value of $s$ ke liye ek. Bada $s$ → steep, jaldi-marti curve. Chhota $s$ → shallow, dhheere-marti curve. $s$ chunna decide karta hai ki tum kaunsa weight use karte ho.

Kabhi kabhi $s$ ko complex number hone diya jaata hai — ek number jiska ek real part aur ek imaginary part hota hai, $s = σ + iω$ likha jaata hai. $sin / cos$ jaise proofs ke liye hum isi par lean karte hain (dekho Euler's Formula $e^{i\theta}$). Jo piece fading control karti hai woh sirf real part hai, jise $Re (s)$ likha jaata hai.

Topic ko yeh kyun chahiye: output $F (s)$ , $s$ ka function hai, aur har convergence condition ( $s > 0$ ya $s > a$ ya $s > ∣ a ∣$ jaise) actually $Re (s)$ ke baare mein ek statement hai ki woh fading race jeetne ke liye kaafi bada ho.

5. $e^{- s t}$ — weight, assembled

Ab §3 aur §4 ko combine karo. Ek $s > 0$ fix karo. Jaise $t$ $0 \to \infty$ chalta hai:

$e^{- s t}_{t = 0} = e^{0} = 1, e^{- s t} t \to \infty 0 (if s > 0) .$

Topic ko yeh kyun chahiye: yeh ek factor hi ek possibly-infinite total ko finite number mein badalta hai. Dono ends par iska behaviour ( $t = 0$ aur $t \to \infty$ ) wahan hai jahan boundary conditions aur convergence strips paida hote hain.

6. $\int_{0}^{\infty} \dots d t$ — "saare forward time par add karo"

Picture: ek curve ke neeche ke region ko hazaron patli vertical rectangles mein katlo, unke saare areas add karo, phir rectangles ko infinitely thin hone do. Exact total integral hai.

Limits $0$ aur $\infty$ ka matlab hai "time zero par add karna shuru karo aur kabhi mat ruko." Jis integral ki upper limit $\infty$ ho use improper integral kehte hain — dekho Improper Integrals.

Topic ko yeh kyun chahiye: transform yahi infinite sum hai. "Kya yeh converge karta hai?" wahi question hai jaise "kya $s$ kaafi bada hai?" — fading weight $e^{- s t}$ ko $f (t)$ ki growth se aage nikalna chahiye.

7. $L {f (t)}$ aur $F (s)$ — machine aur uska output

Picture: ek box jis par $L$ likha hai. Andar $t$ par ek curve jaati hai; bahar $s$ par ek curve aati hai. Dono curves alag alag worlds mein rehti hain — time world aur $s$ -world.

Topic ko yeh kyun chahiye: parent topic ka poora point ek dictionary banana hai — ek do-column table " $t$ ke time world mein $f (t)$ ⟷ $s$ -world mein $F (s)$ ." Har proof us table ki ek row bharta hai.

8. Superpower — yeh machine kyun kisine banayi

Pehle dictionary kyun chahiye: $s$ -world mein koi equation solve karne ke liye tumhe pehchanna hoga ki kaunsa $F (s)$ kaunse $f (t)$ se aaya — aur wapas jaane ke liye reverse lookup chahiye, Inverse Laplace Transform — Partial Fractions. Dono directions ke liye standard table zaroori hai.

9. $n!$ — factorial ( $L {t^{n}}$ ke liye zaroori)

Picture: multiplications ki ek staircase, har step ek naya factor add karti hai. Yeh $L {t^{n}} = n! / s^{n + 1}$ mein aata hai kyunki proof har integration-by-parts step par ek naya factor multiply karta hai (ek recursion).

Topic ko yeh kyun chahiye: yeh woh exact fingerprint hai jo sahi $L {t^{n}} = n! / s^{n + 1}$ ko common galat guess $1/ s^{n + 1}$ se alag karta hai.

10. Do tools jo proofs borrow karte hain

Yeh naye symbols nahi hain balki techniques hain jo proofs assume karti hain ki tum pehle se kar sakte ho:

Integration by parts — $\int u d v = uv - \int v d u$ . $t^{n}$ se ek ek power peelne ke liye, aur $sin \leftrightarrow cos$ swap karne ke liye use hota hai. Full detail Integration by Parts mein.
Euler's formula — $e^{i θ} = cos θ + i sin θ$ . Ek hi exponential integral se dono $sin$ aur $cos$ transforms nikalne ke liye use hota hai. Full detail Euler's Formula $e^{i\theta}$ mein.

Aur do structural facts Linearity and First Shifting Theorem mein prove hue:

Linearity — $L$ sums aur constant multiples ko respect karta hai, toh tum $\frac{e ^{a t} + e ^{- a t}}{2}$ (ek $cosh$ ) ke har piece ko alag transform kar sakte ho.

Yeh foundations topic ko kaise feed karte hain

Ise upar se neeche padho: raw pieces (time, exponential, knob) weighted product assemble karte hain; integral plus convergence use machine $L$ mein wrap karte hain; machine plus teen borrowed tools dictionary produce karte hain; dictionary superpower unlock karti hai.

Equipment checklist

Recall Self-test: kya tum bina dekhe har sawal ka jawab de sakte ho?

Laplace transform mein variable $t$ kis range par hota hai? ::: Zero se $\infty$ tak ka saara time; koi negative time nahi hota. $f (t)$ kya hai, ek sentence mein? ::: Ek rule jo har time $t$ ko ek output number mein badalta hai — time axis ke upar ek curve. Function $e^{x}$ mein kya special hai? ::: Iske growth ki rate uski apni height ke barabar hoti hai; $e^{0} = 1$ , $e^{- x} \to 0$ jaise $x \to \infty$ . $e^{- s t}$ ko "fading weight" kyun kehte hain? ::: $s > 0$ ke liye yeh $1$ se shuru hota hai aur $0$ ki taraf decay karta hai, early times par trust karta hai aur baad walon ko fade out karta hai. Kya $s$ ko integrate kiya jaata hai? ::: Nahi — $s$ ek fixed knob hai jo $t$ par integrate karte waqt constant rehta hai. $Re (s)$ ka matlab kya hai aur yeh kyun matter karta hai? ::: $s$ ka real part; yeh akela control karta hai ki weight fade hota hai ya nahi, isliye saari convergence conditions actually iske baare mein hain. $\int_{0}^{\infty} g d t$ kya compute karta hai? ::: $t = 0$ se $t \to \infty$ tak $g$ ke neeche ka total area (ek improper integral). "Integral converge karta hai" ka matlab kya hai? ::: Jaise upper limit barhti hai, running total infinity ki taraf bhagne ki jagah ek finite number par settle ho jaata hai. $L$ kya hai aur yeh kya output karta hai? ::: Machine " $e^{- s t}$ se multiply karo, $0$ se $\infty$ tak integrate karo"; yeh $s$ ka ek naya function $F (s)$ output karta hai. Hum yeh dictionary banane ki takleef kyun uthate hain? ::: Kyunki $t$ mein differentiate karna $s$ se multiply karna ban jaata hai, ODEs ko algebra mein badalta hai — lekin sirf tab jab hum transforms lookup kar sakein. $0!$ aur $3!$ kya hain? ::: $0! = 1$ aur $3! = 6$ . $sin / cos$ aur $t^{n}$ proofs kaunsi do techniques borrow karti hain? ::: Integration by parts aur Euler's formula.

Connections

Parent: Transform of Derivatives — solving ODEs · Laplace Transform — Definition and Existence
Tools used here: Integration by Parts · Euler's Formula $e^{i\theta}$ · Improper Integrals
Next steps: Linearity and First Shifting Theorem · Inverse Laplace Transform — Partial Fractions