Transforms of standard functions — proofs: Foundations

This page assumes nothing. If you have never seen $\int$ , $e^{- s t}$ , $L$ , $s$ , $\infty$ , or $Re (s)$ , start here and read top-to-bottom. Each symbol is earned before the next one uses it.

0. The whole sentence we are decoding

Everything in the parent topic hangs on this single line:

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

We will now take it apart one symbol at a time, left to right, picture by picture. By the end you should be able to point at every mark on that line and say what it means and what it looks like.

Figure — Transforms of standard functions — proofs

Look at the figure: the blue curve is the raw signal $f (t)$ , the coral curve is the fading weight $e^{- s t}$ , and the shaded mint region is their product — the thing whose area we measure. Keep this picture in mind; every symbol below is one piece of it.

1. $t$ — the input clock

Picture: a number line starting at the point $0$ and marching right forever — $0, 1, 2, \dots$ There is no "before zero" in this story; the machine only cares about $t \geq 0$ .

Why the topic needs it: a Laplace transform is built for signals that "switch on" at a starting moment (a voltage applied at $t = 0$ , a mass released at $t = 0$ ). Restricting to $t \geq 0$ is what makes the lower limit of the integral a clean $0$ .

2. $f (t)$ — the signal, a machine that answers "what is the value at time $t$ ?"

Picture: a curve drawn above the time axis. The height of the curve above the point $t$ is the value $f (t)$ . Examples you will meet: the flat line $f (t) = 1$ , the wiggle $f (t) = sin (t)$ , the runaway growth $f (t) = e^{t}$ .

Why the topic needs it: $f (t)$ is the raw material — the thing being transformed. The entire dictionary of results ( $L {1}$ , $L {t^{n}}$ , $L {sin a t}$ …) is just this same machine fed different curves.

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

Picture: two mirror-image curves.

$e^{x}$ (for $x > 0$ ): a curve that curls upward faster and faster — explosive growth.
$e^{- x}$ (for $x > 0$ ): a curve starting at height $1$ and sliding down toward the floor $0$ , never quite touching it.

Why the topic needs it: the whole trick of the Laplace transform is the fading factor $e^{- s t}$ . Without a decaying weight, adding up a signal over infinite time would usually give infinity. The exponential is what tames that.

4. $s$ — the tuning knob (and why it can be complex)

Picture: think of a family of coral weight-curves, one per value of $s$ . Big $s$ → steep, fast-dying curve. Small $s$ → shallow, slow-dying curve. Choosing $s$ picks which weight you use.

Sometimes $s$ is allowed to be a complex number — a number with a real part and an imaginary part, written $s = σ + iω$ . For proofs like $sin / cos$ we lean on this (see Euler's Formula $e^{i\theta}$). The piece that controls fading is only the real part, written $Re (s)$ .

Why the topic needs it: the output $F (s)$ is a function of $s$ , and every convergence condition (like $s > 0$ or $s > a$ or $s > ∣ a ∣$ ) is really a statement about $Re (s)$ being big enough to win the fading race.

5. $e^{- s t}$ — the weight, assembled

Now combine §3 and §4. Fix an $s > 0$ . As $t$ runs $0 \to \infty$ :

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

Why the topic needs it: this single factor is what turns a possibly-infinite total into a finite number. Its behaviour at the two ends ( $t = 0$ and $t \to \infty$ ) is where boundary conditions and convergence strips are born.

6. $\int_{0}^{\infty} \dots d t$ — "add up over all forward time"

Picture: slice the region under a curve into thousands of skinny vertical rectangles, add all their areas, then let the rectangles get infinitely thin. The exact total is the integral.

The limits $0$ and $\infty$ mean "start adding at time zero and never stop." An integral whose upper limit is $\infty$ is called an improper integral — see Improper Integrals.

Why the topic needs it: the transform is this infinite sum. "Does it converge?" is the same question as "is $s$ big enough?" — the fading weight $e^{- s t}$ must outrun the growth of $f (t)$ .

7. $L {f (t)}$ and $F (s)$ — the machine and its output

Picture: a box labelled $L$ . In goes a curve over $t$ ; out comes a curve over $s$ . The two curves live in different worlds — the time world and the $s$ -world.

Why the topic needs it: the entire point of the parent topic is building a dictionary — a two-column table of " $f (t)$ in the time world ⟷ $F (s)$ in the $s$ -world." Every proof fills in one row of that table.

8. The superpower — why anyone built this machine

Why we need the dictionary first: to solve an equation in the $s$ -world you must recognise which $F (s)$ came from which $f (t)$ — and to go back you need the reverse lookup, Inverse Laplace Transform — Partial Fractions. Both directions require the standard table.

9. $n!$ — the factorial (needed for $L {t^{n}}$ )

Picture: a staircase of multiplications, each step adding one more factor. It appears in $L {t^{n}} = n! / s^{n + 1}$ because the proof multiplies by one new factor at every integration-by-parts step (a recursion).

Why the topic needs it: it is the exact fingerprint that separates the correct $L {t^{n}} = n! / s^{n + 1}$ from the common wrong guess $1/ s^{n + 1}$ .

10. Two tools the proofs borrow

These are not new symbols but techniques the proofs assume you can already run:

Integration by parts — $\int u d v = uv - \int v d u$ . Used to peel a power off $t^{n}$ one at a time, and to swap $sin \leftrightarrow cos$ . Full detail in Integration by Parts.
Euler's formula — $e^{i θ} = cos θ + i sin θ$ . Used to get both the $sin$ and $cos$ transforms from a single exponential integral. Full detail in Euler's Formula $e^{i\theta}$.

And two structural facts proved in Linearity and First Shifting Theorem:

Linearity — $L$ respects sums and constant multiples, so you can transform each piece of $\frac{e ^{a t} + e ^{- a t}}{2}$ (a $cosh$ ) separately.

How these foundations feed the topic

Read it top to bottom: the raw pieces (time, exponential, knob) assemble the weighted product; the integral plus convergence wrap that into the machine $L$ ; the machine plus three borrowed tools produce the dictionary; the dictionary unlocks the superpower.

Equipment checklist

Recall Self-test: can you answer each without peeking?

What does the variable $t$ range over in a Laplace transform? ::: All time from $0$ to $\infty$ ; there is no negative time. What is $f (t)$ , in one sentence? ::: A rule that turns each time $t$ into one output number — a curve above the time axis. What is special about the function $e^{x}$ ? ::: Its rate of growth equals its own height; $e^{0} = 1$ , $e^{- x} \to 0$ as $x \to \infty$ . Why is $e^{- s t}$ called a "fading weight"? ::: For $s > 0$ it starts at $1$ and decays toward $0$ , trusting early times and fading out later ones. Is $s$ integrated over? ::: No — $s$ is a fixed knob held constant while we integrate over $t$ . What does $Re (s)$ mean and why does it matter? ::: The real part of $s$ ; it alone controls whether the weight fades, so all convergence conditions are really about it. What does $\int_{0}^{\infty} g d t$ compute? ::: The total area under $g$ from $t = 0$ to $t \to \infty$ (an improper integral). What does "the integral converges" mean? ::: As the upper limit grows, the running total settles on a finite number instead of running to infinity. What is $L$ and what does it output? ::: The machine "multiply by $e^{- s t}$ , integrate $0$ to $\infty$ "; it outputs a new function $F (s)$ of $s$ . Why do we bother building this dictionary? ::: Because differentiating in $t$ becomes multiplying by $s$ , turning ODEs into algebra — but only if we can look transforms up. What is $0!$ and $3!$ ? ::: $0! = 1$ and $3! = 6$ . Which two techniques do the sin/cos and $t^{n}$ proofs borrow? ::: Integration by parts and 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