Inverse Laplace transform — partial fractions, tables: Foundations

4.6.29 · D1 · Maths › Ordinary Differential Equations › Inverse Laplace transform — partial fractions, tables

Kaatne aur pehchaanne se pehle, tumhe page par har symbol bina rukke padh paana chahiye. Yeh note har ek ko kuch nahi se build karta hai, us order mein jismein ek doosre par depend karte hain. Yahan kuch bhi assumed nahi hai — agar parent note ne use kiya, toh hum define karte hain.

0. Do duniya: $t$ -world aur $s$ -world

Is topic mein sab kuch do "duniyon" ke beech conversation hai.

$t$ -world woh jagah hai jahan physics rehti hai. Yahan $t$ ka matlab time hai, ek real number jo $0$ se start hota hai aur badhta hai. Ek function $f (t)$ ek picture hai: ek height jo time axis ke saath daayein slide karte waqt badlti hai.
$s$ -world ek bookkeeping duniya hai. Yahan $s$ ek variable hai jo humne banaya algebra ko easy karne ke liye. Is course mein hum $s$ ko ek real number maante hain; advanced treatments mein $s$ ko complex hone diya jaata hai (ek real part plus imaginary part). Ek function $F (s)$ ek alag picture hai — ek height jo $s$ axis ke saath slide karte waqt badlti hai. Yeh $f (t)$ ki same information carry karta hai, bas re-packaged.

Figure — Inverse Laplace transform — partial fractions, tables

Topic ko yeh kyun chahiye: forward Laplace transform ek $f (t)$ leta hai aur ek $F (s)$ produce karta hai. Inverse doosri taraf jaata hai. Agar tumhare dimag mein "do duniya" nahi hai, toh tum ek $t$ -answer ko ek $s$ -expression se confuse kar loge — panic ka sabse common source.

1. Function machine notation $f (t)$ aur $F (s)$

$f (t)$ ko zor se padho " $f$ of $t$ ": $f$ naama ek machine jo ek number $t$ khaati hai aur ek number ugalti hai. Naam lowercase hai kyunki yeh $t$ -world mein rehta hai.

$F (s)$ ek alag machine hai, capital $F$ se named, $s$ -world mein rehti hai. Is poore topic mein convention:

Topic ko yeh kyun chahiye: parent note mein har theorem ek pairing ke roop mein stated hai — "agar $L^{- 1} {F} = f$ , toh …". Lowercase/CAPITAL convention woh shorthand hai jo tumhe ek nazar mein batata hai ki rule ka kaunsa side answer-world ( $t$ ) mein hai aur kaunsa algebra-world ( $s$ ) mein. Yeh convention miss karo aur tum shift theorems ko parse bhi nahi kar paoge.

2. $\int_{0}^{\infty} \dots d t$ — integral sign, zero se

Forward transform ek integral ke saath likha jaata hai. Tumhe ise padh paana chahiye bhalehe hum yahan ise rarely compute karte hain.

Neeche ka $0$ aur upar ka $\infty$ limits hain: $t = 0$ par add karna shuru karo, forever chalte raho.
$g (t) d t$ ek strip ka area hai = (height $g (t)$ ) $\times$ (width $d t$ ).

Topic ko yeh kyun chahiye: definition $F (s) = \int_{0}^{\infty} e^{- s t} f (t) d t$ har table entry ka birth certificate hai, aur har entry sirf apne ROC ke andar valid hai. Tum invert karte waqt yeh integral nahi karte — lekin tumhe trust karna hota hai ki yeh exist karta hai (sahi $s$ par) aur har $f (t)$ ke liye ek single, unique $F (s)$ deta hai. Laplace Transform — definition and existence dekho.

3. $e$ , $e^{a t}$ , aur $e^{- s t}$ — growth/decay machine

Letter $e \approx 2.718$ ek fixed number hai, jaise $π$ . $e^{a t}$ ko special banane wali baat ek picture hai.

$a$ ke sign ke har case ko cover karo:

$a$ ka sign	picture	naam
$a > 0$	teji se aur teji se badhta hai	growth
$a = 0$	height $1$ par horizontal line	constant
$a < 0$	girta hai, $0$ ki taraf flatten hota hai	decay

Transform integral ke andar factor $e^{- s t}$ ek decay weight hai — lekin sirf tab jab $s > 0$ . Agar $s \leq 0$ toh yeh decay karna band kar deta hai, jo exactly Section 2 ki divergence warning hai. Jab yeh decay karta hai, yahi woh cheez hai jo infinite-area integral ko ek finite number par settle karati hai.

Topic ko yeh kyun chahiye: pehli table row hi $L^{- 1} {1/ (s - a)} = e^{a t}$ hai. Ek pole $\frac{1}{s - a}$ ko instantly "rate $a$ wala ek exponential" padhna woh reflex hai jis par poora method bana hai. Do exponentials kaise $sinh, cosh$ mein combine hote hain yeh dekhne ke liye Hyperbolic functions sinh/cosh vs sin/cos dekho.

4. $L$ aur $L^{- 1}$ — transform arrows

$L$ ek fancy script L hai. $L {f (t)}$ ko padho " $f$ ka Laplace transform". Yeh ek arrow hai jo $t$ -world se $s$ -world ki taraf point karta hai.

$L^{- 1}$ (chhote $- 1$ ke saath) reverse arrow hai, $s$ -world se wapas $t$ -world ki taraf. $- 1$ ka matlab "ek upar" nahi hai; iska matlab hai "undo", exactly jaise U-turn ek turn ko undo karta hai.

Do arrows cancel ho jaate hain: $L^{- 1} {L {f}} = f$ . Woh round trip hi poora point hai.

Topic ko yeh kyun chahiye: poora chapter ek round trip hai. Tum $L$ use karte ho ek differential equation ko algebra mein turn karne ke liye, algebra solve karte ho, phir $L^{- 1}$ use karte ho — is topic ka star — answer ghar laane ke liye. Explicit "undo" arrow ke bina us operation ka koi naam nahi hai jo note sikhata hai, aur " $- 1$ ≠ reciprocal" reflex line one par ek fatal algebra slip rokta hai.

5. "Undoing" ek unique answer kyun deta hai — Lerch

Agar do alag $f (t)$ ek hi $F (s)$ produce kar sakti, toh lookup table useless hoti (kaunsi likhte ho?). Khush-qismati se, un functions ke liye jo hum milte hain (nice, exponential se zyada fast explode nahi karti — yaani of exponential order, exactly upar wali ROC condition), forward arrow one-to-one hai: alag inputs alag outputs dete hain. Toh reverse arrow exactly ek $f (t)$ par land karta hai. Yeh guarantee Lerch's theorem hai, aur yeh woh silent permission slip hai jo humein "table mein lookup karne" deti hai.

Exact conditions ke liye Laplace Transform — definition and existence dekho (piecewise continuous, of exponential order).

6. "Linear" — woh property jo humein split karne deti hai

Ise ek sorting machine ki tarah socho jo kabhi items mix nahi karti: agar tum ek mixture daalo, har ingredient apne aap inverted ho kar aata hai, apne number se scaled, aur tum bas results add karte ho. Yahi partial fractions ka license hai: hum allowed hain $F (s)$ ko ek sum mein chop karne aur har chunk ko independently invert karne ke liye. Linearity ke bina "chop and glue" strategy illegal hoti.

7. Rational functions $\frac{P ( s )}{Q ( s )}$ — upar, neeche, degree

Almost har $F (s)$ jo tum invert karte ho ek rational function hota hai: ek polynomial doosre se divided.

$P (s)$ = upar (numerator).
$Q (s)$ = neeche (denominator).
Degree = $s$ ki sabse badi power jo present hai. $3 s^{2} - 7$ ki degree $2$ hai.
Factor = ek piece jis se neeche multiply out hoti hai. Jaise $12 = 2 \times 2 \times 3$ , polynomial $s^{2} - 1 = (s - 1) (s + 1)$ ke factors $(s - 1)$ aur $(s + 1)$ hain. $Q (s)$ ko factor karna matlab ise aisa product likhna.
Pole = $s$ ki woh value jahan neeche zero ho jaata hai (fraction blow up karta hai). Poles $Q (s)$ ke factors se aate hain.

Topic ko yeh kyun chahiye: partial fractions $F (s)$ ko $Q (s)$ ke factors ke basis par split karta hai — distinct linear, repeated linear, ya irreducible quadratic. Tum classify nahi kar sakte jo tum naam nahi de sakte (upar, neeche, degree, factor, pole).

8. $ω$ , $sin ω t$ , $cos ω t$ — wobble machines

$ω$ (Greek "omega") ek positive number hai jise angular frequency kehte hain: ek wave kitni fast wobble karti hai. $sin ω t$ aur $cos ω t$ do basic waves hain jo upar-neeche jaati rehti hain forever, kabhi fade nahi hoti.

Table ke liye key reading reflex:

$\frac{s}{s ^{2} + ω ^{2}} \to cos ω t$ ( $t = 0$ par high start, cosine ki tarah $1$ par).
$\frac{ω}{s ^{2} + ω ^{2}} \to sin ω t$ ( $0$ par start, sine ki tarah).

Jab yeh decaying $e^{a t}$ ke saath combine hote hain, tumhe ek damped wobble milta hai (energy khota spring). Woh pairing exactly First and Second Shift Theorems engine hai.

9. "Complete the square" — bottom ko reshape karna

Jab $Q (s)$ mein irreducible quadratic ho (Section 7), tum ise $(s - a) (s - b)$ mein factor nahi kar sakte. Iske bajaaye tum complete the square karte ho ise $(s - a)^{2} + ω^{2}$ shape mein force karne ke liye, exactly woh form jo shift table ko chahiye. Example: $s^{2} + 2 s + 5 = (s + 1)^{2} + 4$ , toh $a = - 1$ aur $ω = 4 = 2$ .

Prerequisite map

Arrows ko upar parent tak follow karo: the main topic note sabse upar hai; is page ka sab kuch usme feed karta hai. Wahan se natural next steps hain Solving ODEs with Laplace Transforms aur Convolution Theorem; Heaviside Step & Dirac Delta switching-on functions add karta hai.

Equipment checklist

Khud test karo — har ::: line ka right side cover karo aur zor se jawab do.

Do "worlds" kya refer karti hain?

t

-world (physical time, functions

f (t)

) aur

s

-world (algebra bookkeeping, functions

F (s)

Kya

s

real hai ya complex?

is course mein real; generally complex (

s = σ + iω

f (t)

F (s)

kaise padhte ho?

lowercase = ek

t

-world function; capital = uska transformed

s

-world partner.

\int_{0}^{\infty} g (t) d t

ek picture mein kya compute karta hai?

t = 0

t = \infty

tak

g

ke neeche total area, yaani infinitely many thin strips ka sum.

Transform integral diverge kyun ho sakta hai, aur kya fix karta hai?

agar

s \leq 0

toh weight

e^{- s t}

decay nahi karta; humein

s > σ_{0}

chahiye (region of convergence) taaki integral converge kare.

a > 0

a = 0

a < 0

ke liye

e^{a t}

ki picture kya hai?

snowballing growth,

1

par flat line,

0

ki taraf flatten hota decay.

L^{- 1}

mein

- 1

ka kya matlab hai?

"transform undo karo" — reverse arrow, na ki

L

par one over.

Inverse ek single well-defined answer kyun hai?

Lerch's theorem — forward transform nice functions par one-to-one hai.

L^{- 1}

ki linearity state karo.

L^{- 1} {a F + b G} = a L^{- 1} {F} + b L^{- 1} {G}

Kaise test karte ho ki ek quadratic irreducible hai?

s^{2} + b s + c

ka discriminant

b^{2} - 4 c

check karo; irreducible iff yeh

< 0

hai (koi real roots nahi).

P (s) / Q (s)

proper kab hota hai?

jab

de g P < de g Q

; warna pehle long-divide karo.

\frac{∙}{s ^{2} + ω ^{2}}

mein,

s

-on-top kya deta hai,

ω

-on-top kya deta hai?

s

-on-top →

cos ω t

;

ω

-on-top →

sin ω t

(s + 1)^{2} + 4

a

aur

ω

kya hain?

a = - 1

aur

ω = 4 = 2