Laplace transform — definition, region of convergence: Foundations

Before you can read the parent note (Laplace transform — definition, ROC) with zero confusion, you must own every symbol it fires at you. We build them one at a time, each from the one before.

0. The characters, in order of appearance

Here is the full cast the parent note uses. We meet each below.

Symbol	Read as	Section
$t$	time	§1
$f (t)$	a function of time	§2
$\int_{a}^{b} \dots d t$	"add up over $t$ "	§3
$\int_{0}^{\infty}$	improper integral	§4
$e^{x}$ , $e^{- s t}$	the exponential / kernel	§5
$s = σ + iω$	the complex frequency knob	§6
$Re (s) = σ$	the real part	§6
$L {f} = F (s)$	the transform itself	§7
"converges", ROC, abscissa $a$	when the sum settles	§8
exponential order, $\leq M e^{a t}$	growth speed limit	§9

1. $t$ — time, the input axis

Picture: a horizontal number line pointing right. $t = 0$ is "now"; everything to the right is later.

Why the topic needs it: the signals we transform (a voltage, a temperature, a swinging pendulum) are things that change as time passes. $t$ is the axis they live on.

2. $f (t)$ — a function of time

Picture: a curve drawn above the time line. For each $t$ on the floor, the curve's height is $f (t)$ . See the figure: three signals — a flat one ( $f = 1$ ), a rising one ( $f = t$ ), and a wiggling one ( $sin$ ).

Figure — Laplace transform — definition, region of convergence

Why the topic needs it: $f (t)$ is the raw material — the thing going into the transform machine. Every example in the parent ( $1$ , $t$ , $e^{a t}$ , $sin ω t$ ) is a specific choice of $f (t)$ .

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

Picture: the curve $g (t)$ with the region beneath it shaded and chopped into rectangles.

$\int$ is a stretched "S" for Sum.
$d t$ is the tiny width of one slice.
$a$ (bottom) and $b$ (top) are the limits — where the adding starts and stops.

Why the topic needs it: the Laplace transform is an integral. It adds up a whole signal into a single number for each $s$ . If you can't picture "area = sum of strips," the definition is just squiggles.

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

Picture: the shaded area keeps extending rightward forever. Two things can happen:

the running total settles toward a fixed height (we say it converges), or
it runs off to infinity (it diverges).

Why the topic needs it: the Laplace integral runs to $\infty$ . Whether it converges is the entire drama of the Region of Convergence. See Improper Integrals for the full machinery.

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

Picture: two curves — one sliding down to the floor (decay), one rocketing up (growth). See the figure.

Why the topic needs it: $e^{- s t}$ is the heart of the transform — the "fade-out filter." Everything about the ROC comes from a race between $e^{- s t}$ fading and $f (t)$ growing.

6. $s = σ + iω$ and its real part $Re (s)$

Picture: a 2D plane (the " $s$ -plane"). Horizontal axis $= σ$ , vertical axis $= ω$ . Each choice of $s$ is a point on this plane.

The one fact that governs convergence: the size of $e^{- s t}$ depends only on $σ$ , not on $ω$ : $∣ e^{- s t} ∣ = e^{- (σ + iω) t} = e^{- σ t} .$ The $iω$ part only spins (rotates), it never changes magnitude. So decay is controlled by $σ = Re (s)$ alone.

Why the topic needs it: the ROC is a region in the $s$ -plane — specifically a half-plane cut by a vertical line $σ = a$ . You cannot picture that region without the $s$ -plane.

7. $L {f} = F (s)$ — the transform

Picture: a box labeled $L$ . In goes a curve $f (t)$ over time; out comes a curve $F (s)$ over the $s$ -plane. Same information, new language.

Why the topic needs it: this is literally the definition the parent builds. Everything before was scaffolding so this line reads as plain English: "fade $f$ with $e^{- s t}$ , add it all up over time, and record the total as a function of the fade-speed $s$ ." Going back is Inverse Laplace Transform.

8. Convergence, ROC, and the abscissa $a$

Picture: the $s$ -plane with a vertical dividing line at $σ = a$ . Everything to its right is safe (converges); everything left diverges.

Why: if $f$ grows like $e^{a t}$ , the integrand is $e^{- σ t} e^{a t} = e^{(a - σ) t}$ . This decays only when $a - σ < 0$ , i.e. $σ > a$ . The faster $f$ grows, the further right you must slide $s$ .

9. Exponential order — the growth speed limit

Picture: the curve $∣ f (t) ∣$ eventually trapped underneath a ceiling $M e^{a t}$ . As long as $f$ stays below some exponential, it has a Laplace transform.

Why the topic needs it: this is the sufficient condition for the transform to exist. A monster like $e^{t^{2}}$ beats every $e^{a t}$ — it pokes through every ceiling — so it has no Laplace transform at all. See Exponential Order and Growth Rates.

Prerequisite map

Read it bottom-out: time and functions give us something to integrate; the exponential builds the kernel; complex $s$ supplies the fade-speed and (via its real part) decides decay; together they produce the transform, and the growth-vs-decay race carves out the ROC.

Equipment checklist

Cover the right side; can you answer before revealing?

What does

t \geq 0

restrict us to?

only the future — signals starting at "now,"

t = 0

f (t)

in one phrase?

a rule giving one output value for each moment of time.

What does

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

compute, pictorially?

the area under

g

, as a sum of skinny strips of width

d t

Why is

\int_{0}^{\infty}

called improper?

the top limit is infinite; we take

lim_{T \to \infty}

of the finite area.

The two outcomes of an improper integral?

it converges (settles to a finite number) or diverges (runs to infinity).

The defining superpower of

e^{x}

its derivative equals itself,

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

Does

e^{- k t}

(with

k > 0

) grow or decay?

decay — it slides from

1

down toward

0

Two reasons

e^{- s t}

is the kernel?

it tames growth (decays), and

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

turns

d / d t

into

\times s

Write

s

in real/imaginary parts.

s = σ + iω

, with

σ = Re (s)

Which part of

s

controls convergence, and why?

σ = Re (s)

, because

∣ e^{- s t} ∣ = e^{- σ t}

(the

iω

only rotates).

What is

L {f}

in words?

fade

f

e^{- s t}

, add it all over time, record the total as

F (s)

Shape of every ROC?

a right half-plane

Re (s) > a

What is the abscissa of convergence

a

the boundary value; it equals the signal's growth rate.

f

is of exponential order

a

when...?

∣ f (t) ∣ \leq M e^{a t}

for all

t \geq T

, some

M > 0

T \geq 0

Why does

e^{t^{2}}

fail?

it outgrows every

e^{a t}

, so no kernel can tame it — transform never converges.

Connections

Parent topic (Hinglish)
Improper Integrals — the convergence machinery behind §4 and the ROC.
Exponential Order and Growth Rates — the growth-speed idea of §9.
Inverse Laplace Transform — undoing the machine of §7.
Laplace Transform of Derivatives — cashes in the $\times s$ property of §5.
Solving ODEs with Laplace Transforms — the payoff.
Fourier Transform — the special case $Re (s) = 0$ .

#flashcard