Foundations — Inverse Laplace transform — partial fractions, tables

Before you can chop and recognise, you must be able to read every symbol on the page without pausing. This note builds each one from nothing, in the order they lean on each other. Nothing here is assumed — if the parent note used it, we define it.

0. Two worlds: the $t$-world and the $s$-world

Everything in this topic is a conversation between two "worlds".

• The $t$-world is where physics lives. Here $t$ means time, a real number that starts at $0$ and grows. A function $f(t)$ is a picture: a height that changes as you slide right along the time axis.
• The $s$-world is a bookkeeping world. Here $s$ is a variable we invented to make algebra easy. In this course we treat $s$ as a real number; in more advanced treatments $s$ is allowed to be complex (a real part plus an imaginary part). A function $F(s)$ is a different picture — a height that changes as you slide along the $s$ axis. It carries the same information as $f(t)$, only re-packaged.

Why the topic needs this: the forward Laplace transform takes an $f(t)$ and produces an $F(s)$. The inverse goes the other way. If you don't hold "two worlds" in your head, you'll confuse a $t$-answer with an $s$-expression — the single most common source of panic.

1. The function machine notation $f(t)$ and $F(s)$

Read $f(t)$ out loud as "$f$ of $t$": a machine named $f$ that eats a number $t$ and spits out a number. The name is lowercase because it lives in the $t$-world.

$F(s)$ is a different machine, named with a capital $F$, living in the $s$-world. The convention across this whole topic:

Why the topic needs this: every theorem in the parent note is stated as a pairing — "if ${\mathcal{L}}^{-1}\{F\}=f$, then …". The lowercase/CAPITAL convention is the shorthand that tells you at a glance which side of a rule lives in the answer-world ($t$) and which lives in the algebra-world ($s$). Miss the convention and you cannot even parse the shift theorems.

2. ${\int }_0^{\infty }\dots dt$ — the integral sign, from zero

The forward transform is written with an integral. You must be able to read it even though we rarely compute it here.

• The $0$ at the bottom and $\infty$ at the top are the limits: start adding at $t=0$, keep going forever.
• $g(t)dt$ is one strip's area = (height $g(t)$) $\times$ (width $dt$).

Why the topic needs it: the definition $F(s)={\int }_0^{\infty }{e}^{-st}f(t)dt$ is the birth certificate of every table entry, and each entry is only valid inside its ROC. You don't do this integral when inverting — but you must trust it exists (on the right $s$) and gives a single, unique $F(s)$ for each $f(t)$. See Laplace Transform — definition and existence.

3. $e$, $e^{at}$, and $e^{-st}$ — the growth/decay machine

The letter $e\approx 2.718$ is a fixed number, like $\pi$. What makes $e^{at}$ special is a picture.

Cover every case of the sign of $a$:

The factor $e^{-st}$ inside the transform integral is a decay weight — but only when $s>0$. If $s\le 0$ it stops decaying, which is exactly the divergence warning from Section 2. When it does decay, it is what makes the infinite-area integral settle to a finite number.

Why the topic needs it: the very first table row is ${\mathcal{L}}^{-1}\{1/(s-a)\}=e^{at}$. Reading a pole $\frac{1}{s-a}$ instantly as "an exponential with rate $a$" is the reflex the whole method is built on. See Hyperbolic functions sinh/cosh vs sin/cos for how two exponentials combine into $\sinh ,\cosh$.

4. $\mathcal{L}$ and ${\mathcal{L}}^{-1}$ — the transform arrows

$\mathcal{L}$ is a fancy script L. Read $\mathcal{L}\{f(t)\}$ as "the Laplace transform of $f$". It is an arrow pointing from the $t$-world to the $s$-world.

${\mathcal{L}}^{-1}$ (with the little $-1$) is the reverse arrow, from the $s$-world back to the $t$-world. The $-1$ does not mean "one over"; it means "undo", exactly like a U-turn undoes a turn.

The two arrows cancel: ${\mathcal{L}}^{-1}\{\mathcal{L}\{f\}\}=f$. That round trip is the whole point.

Why the topic needs this: the entire chapter is one round trip. You use $\mathcal{L}$ to turn a differential equation into algebra, solve the algebra, then use ${\mathcal{L}}^{-1}$ — the star of this topic — to bring the answer home. Without the explicit "undo" arrow there is no name for the very operation the note teaches, and the "$-1$ ≠ reciprocal" reflex prevents a fatal algebra slip on line one.

5. Why "undoing" gives a unique answer — Lerch

If two different $f(t)$'s could produce the same $F(s)$, a lookup table would be useless (which one do you write down?). Luckily, for the functions we meet (nice, not exploding faster than an exponential — i.e. of exponential order, exactly the ROC condition above), the forward arrow is one-to-one: different inputs give different outputs. So the reverse arrow lands on exactly one $f(t)$. This guarantee is Lerch's theorem, and it is the silent permission slip that lets us "look it up in a table".

See Laplace Transform — definition and existence for the exact conditions (piecewise continuous, of exponential order).

6. "Linear" — the property that lets us split

Picture it as a sorting machine that never mixes items: if you pour in a mixture, each ingredient comes out inverted on its own, scaled by its own number, and you just add the results. This is the licence for partial fractions: we are allowed to chop $F(s)$ into a sum and invert each chunk independently. Without linearity the "chop and glue" strategy would be illegal.

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

Almost every $F(s)$ you invert is a rational function: one polynomial divided by another.

• $P(s)$ = the top (numerator).
• $Q(s)$ = the bottom (denominator).
• Degree = the highest power of $s$ present. Degree of $3s^2-7$ is $2$.
• A factor = a piece the bottom is multiplied out of. Just as $12=2\times 2\times 3$, the polynomial $s^2-1=(s-1)(s+1)$ has factors $(s-1)$ and $(s+1)$. Factoring $Q(s)$ means writing it as such a product.
• A pole = a value of $s$ where the bottom hits zero (the fraction blows up). Poles come from the factors of $Q(s)$.

Why the topic needs it: partial fractions splits $F(s)$ based on the factors of $Q(s)$ — distinct linear, repeated linear, or irreducible quadratic. You cannot classify what you cannot name (top, bottom, degree, factor, pole).

8. $\omega$, $\sin \omega t$, $\cos \omega t$ — the wobble machines

$\omega$ (Greek "omega") is a positive number called the angular frequency: how fast a wave wobbles. $\sin \omega t$ and $\cos \omega t$ are the two basic waves that go up and down forever, never fading.

The key reading reflex for the table:

• $\frac{s}{s^2+{\omega }^2}\to \cos \omega t$ (starts high at $t=0$, like a cosine at $1$).
• $\frac{\omega }{s^2+{\omega }^2}\to \sin \omega t$ (starts at $0$, like a sine).

When these combine with a decaying $e^{at}$, you get a damped wobble (a spring losing energy). That pairing is exactly the First and Second Shift Theorems engine.

9. "Complete the square" — reshaping a bottom

When $Q(s)$ has an irreducible quadratic (Section 7), you cannot factor it into $(s-a)(s-b)$. Instead you complete the square to force it into the shape $(s-a{)}^2+{\omega }^2$, the exact form the shift table needs. Example: $s^2+2s+5=(s+1{)}^2+4$, so $a=-1$ and $\omega =\sqrt{4}=2$.

The prerequisite map

Follow the arrows up to the parent: the main topic note sits at the top; everything on this page feeds into it. From there the natural next steps are Solving ODEs with Laplace Transforms and Convolution Theorem; Heaviside Step & Dirac Delta adds the switching-on functions.

Equipment checklist

Test yourself — cover the right side of each ::: line and answer aloud.