Two-sided Laplace transform

Key takeaways

Two-sided Laplace transform belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Two-sided Laplace transform to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Two-sided Laplace transform from memory before moving on to harder problems.

Reference excerpt

In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform. If ⁠ f ( t ) {\displaystyle f(t)} ⁠ is a real- or complex-valued function of the real variable ⁠ t {\displaystyle t} ⁠ defined for all real numbers, then the two-sided Laplace transform is defined by the integral

B { f } ( s ) = F ( s ) = ∫ − ∞ ∞ e − s t f ( t ) d t . {\displaystyle {\mathcal {B}}\{f\}(s)=F(s)=\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.}

The integral is most commonly understood as an improper integral, which converges if and only if both integrals

∫ 0 ∞ e − s t f ( t ) d t , ∫ − ∞ 0 e − s t f ( t ) d t {\displaystyle \int _{0}^{\infty }e^{-st}f(t)\,dt,\quad \int _{-\infty }^{0}e^{-st}f(t)\,dt}

exist. There seems to be no generally accepted notation for the two-sided transform; the

B {\displaystyle {\mathcal {B}}} used here recalls "bilateral". The two-sided transform used by some authors is

T { f } ( s ) = s B { f } ( s ) = s F ( s ) = s ∫ − ∞ ∞ e − s t f ( t ) d t . {\displaystyle {\mathcal {T}}\{f\}(s)=s{\mathcal {B}}\{f\}(s)=sF(s)=s\int _{-\infty }^{\infty }e^{-st}f(t)\,dt.}

In pure mathematics the argument ⁠ t {\displaystyle t} ⁠ can be any variable, and Laplace transforms are used to study how differential operators transform the function. In science and engineering applications, the argument ⁠ t {\displaystyle t} ⁠ often represents time (in seconds), and the function ⁠ f ( t ) {\displaystyle f(t)} ⁠ often represents a signal or waveform that varies with time. In these cases, the signals are transformed by filters, that work like a mathematical operator, but with a restriction. They have to be causal, which means that the output at a given time ⁠ t {\displaystyle t} ⁠ cannot depend on an input that occurs at later time ⁠ t ′ {\displaystyle t'} ⁠. In population ecology, the argument ⁠ t {\displaystyle t} ⁠ often represents spatial displacement in a dispersal kernel. When working with functions of time, ⁠ f ( t ) {\displaystyle f(t)} ⁠ is called the time domain representation of the signal, while ⁠ F ( s ) {\displaystyle F(s)} ⁠ is called the s-domain (or Laplace domain) representation. The inverse transformation then represents a synthesis of the signal as the sum of its frequency components taken over all frequencies, whereas the forward transformation represents the analysis of the signal into its frequency components.

Relationship to the Fourier transform The Fourier transform can be defined in terms of the two-sided Laplace transform ⁠ F {\displaystyle F} ⁠ of a function ⁠ f {\displaystyle f} ⁠:

F { f ( t ) } = F ( s ) | s = i ω = F ( i ω ) . {\displaystyle {\mathcal {F}}\{f(t)\}=\left.F(s)\right|_{s=i\omega }=F(i\omega ).}

Note that definitions of the Fourier transform differ, and in particular

F { f ( t ) } = F ( s ) | s = i ω = 1 2 π B { f ( t ) } ( s ) {\displaystyle {\mathcal {F}}\{f(t)\}=\left.F(s)\right|_{s=i\omega }={\frac {1}{\sqrt {2\pi }}}{\mathcal {B}}\{f(t)\}(s)}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Two-sided Laplace transform

Start with the simplest possible case. Write down what Two-sided Laplace transform claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Two-sided Laplace transform before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Two-sided Laplace transform ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Two-sided Laplace transform

In research: Two-sided Laplace transform appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Two-sided Laplace transform in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Two-sided Laplace transform is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integral transforms, Laplace transforms, so understanding it makes those chapters shorter.
In everyday life: Look for Two-sided Laplace transform outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Two-sided Laplace transform in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Two-sided Laplace transform means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Two-sided Laplace transform out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Two-sided Laplace transform in simple terms?

In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-si…

Why does Two-sided Laplace transform matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Two-sided Laplace transform?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Two-sided Laplace transform.