Laplace transform
E139491
The Laplace transform is an integral transform widely used in mathematics, physics, and engineering to convert functions of time into functions of a complex variable, simplifying the analysis and solution of differential equations and linear systems.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Laplace transform canonical | 9 |
| Laplace–Borel transform | 1 |
| one-sided Laplace transform | 1 |
| two-sided Laplace transform | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1221702 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Laplace transform Context triple: [Pierre-Simon Laplace, developedConcept, Laplace transform]
-
A.
Fourier analysis
Fourier analysis is a mathematical method for decomposing functions or signals into sums of sinusoidal components, widely used in fields such as signal processing, physics, and engineering.
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B.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
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C.
Fourier
Fourier is a French surname most famously associated with Jean-Baptiste Joseph Fourier, the mathematician and physicist known for developing Fourier analysis and Fourier series.
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D.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
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E.
Bode plot
A Bode plot is a graphical representation of a linear system’s frequency response, showing magnitude and phase versus frequency on logarithmic scales, widely used in control and amplifier design.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Laplace transform Target entity description: The Laplace transform is an integral transform widely used in mathematics, physics, and engineering to convert functions of time into functions of a complex variable, simplifying the analysis and solution of differential equations and linear systems.
-
A.
Fourier analysis
Fourier analysis is a mathematical method for decomposing functions or signals into sums of sinusoidal components, widely used in fields such as signal processing, physics, and engineering.
-
B.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
-
C.
Fourier
Fourier is a French surname most famously associated with Jean-Baptiste Joseph Fourier, the mathematician and physicist known for developing Fourier analysis and Fourier series.
-
D.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
E.
Bode plot
A Bode plot is a graphical representation of a linear system’s frequency response, showing magnitude and phase versus frequency on logarithmic scales, widely used in control and amplifier design.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
integral transform
ⓘ
mathematical concept ⓘ |
| appliedIn |
control theory
ⓘ
heat conduction problems ⓘ probability theory ⓘ queueing theory ⓘ systems engineering ⓘ vibration analysis ⓘ |
| commonPair |
L{1} = 1/s
ⓘ
L{cos(bt)} = s/(s^2 + b^2) ⓘ L{e^{at}} = 1/(s−a) ⓘ L{sin(bt)} = b/(s^2 + b^2) ⓘ L{t^n} = n!/s^{n+1} ⓘ L{u(t)} = 1/s ⓘ L{δ(t)} = 1 ⓘ |
| convolutionProperty | L{(f*g)(t)} = F(s) G(s) ⓘ |
| differentiationProperty | L{f'(t)} = s F(s) − f(0⁺) ⓘ |
| domain | time domain to complex frequency domain ⓘ |
| field |
engineering
ⓘ
mathematics ⓘ physics ⓘ |
| finalValueTheorem | lim_{t→∞} f(t) = lim_{s→0} s F(s) ⓘ |
| frequencyShiftProperty | L{e^{at} f(t)} = F(s−a) ⓘ |
| generalizationOf |
one-sided Laplace transform
ⓘ
linked to:
Laplace transform
|
| hasInverse | inverse Laplace transform ⓘ |
| hasVariant |
two-sided Laplace transform
ⓘ
linked to:
Laplace transform
|
| initialValueTheorem | f(0⁺) = lim_{s→∞} s F(s) ⓘ |
| integrationProperty | L{∫₀^t f(τ) dτ} = F(s)/s ⓘ |
| inverseDefinition | f(t) = (1/(2πi)) ∫_{γ−i∞}^{γ+i∞} F(s) e^{st} ds ⓘ |
| kernel | e^{-st} ⓘ |
| linearity | L{a f(t) + b g(t)} = a L{f(t)} + b L{g(t)} ⓘ |
| mapsFrom | functions of a real variable ⓘ |
| mapsTo | functions of a complex variable ⓘ |
| namedAfter | Pierre-Simon Laplace ⓘ |
| nthDerivativeProperty | L{f^{(n)}(t)} = s^n F(s) − s^{n−1} f(0⁺) − … − f^{(n−1)}(0⁺) ⓘ |
| regionOfConvergence | set of complex s where the integral converges ⓘ |
| relatedTransform |
Fourier transform
ⓘ
Z-transform ⓘ |
| simplifies |
handling of initial conditions in differential equations
ⓘ
solution of linear ordinary differential equations with constant coefficients ⓘ |
| standardDefinition | L{f(t)}(s) = ∫₀^∞ e^{-st} f(t) dt ⓘ |
| timeShiftProperty | L{f(t−a)u(t−a)} = e^{-as} F(s) ⓘ |
| transformVariableName | s ⓘ |
| usedFor |
analyzing linear time-invariant systems
ⓘ
circuit analysis ⓘ control system analysis ⓘ signal processing ⓘ solving linear differential equations ⓘ stability analysis ⓘ |
| variableName | t ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Laplace transform Description of subject: The Laplace transform is an integral transform widely used in mathematics, physics, and engineering to convert functions of time into functions of a complex variable, simplifying the analysis and solution of differential equations and linear systems.
Referenced by (12)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Laplace transform
linked to: Laplace transform
linked to: Laplace transform
subject linked to:
Mellin transforms
subject linked to:
Volterra integral equations
subject linked to:
"Signals and Systems"