# Laplace transform <!-- MICROSIMGEN:BEGIN v1.7 — generated by g08_place_microsims.py; three.js first (§15); do not hand-edit inside --> ## Microsims — p5.js ### Laplace transform (p5.js) · `∫ x e^{−st}` <div class="microsim-player"> <iframe src="https://editor.p5js.org/sciencenibber/full/nG5JCk11H" width="100%" height="480" frameborder="0" loading="lazy" sandbox="allow-scripts allow-same-origin" title="Laplace transform — p5.js microsim"></iframe> </div> *Fourier's cousin with a decay axis; poles and zeros in the s-plane make stability visible.* **Open in the editor:** [&#9654; fork this sketch](https://editor.p5js.org/sciencenibber/sketches/nG5JCk11H) · movement *II · Transforms & the frequency domain* · library `p5js` ### Related microsims Live sims on neighbouring articles — 6 of them inside this article's own Wikipedia link tree: - [[Complex_analysis]] *(in tree)* - [[Control_theory]] *(in tree)* - [[Convolution]] *(in tree)* - [[Frequency_domain]] *(in tree)* - [[Impulse_response]] *(in tree)* - [[Integral_transform]] *(in tree)* *Sim hosted off-article; the article owns the reference, not the runtime (WIKI_RULES §10.4). 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See it next [](Integral_transform) *→ [[Integral_transform|Integral transform]]* <!-- VISUAL-LINK:END --> --- Back to Signal Processing Portal · the room · Semiotic gateway <!-- REAL-GENERATIVE-MEDIA:START --> ## What it is The Laplace transform maps a time function f(t) to a function of a complex variable via F(s) = ∫₀^∞ f(t) e^{−st} dt, where s = σ + jω is the complex frequency. ## How it works / why it matters It generalizes the Fourier transform by adding a real exponential factor, so it converges for a wider class of signals and encodes both oscillation (jω) and growth or decay (σ). Differentiation and convolution in time become multiplication in the s-domain, turning linear differential equations into algebra; the resulting transfer functions, poles, and zeros are the standard language of continuous-time system and control analysis. The inverse transform recovers the time-domain response. ## Signs & universals Instantiates: transformation, frequency, signal. ## Related [[Integral_transform]] · Fourier transform · [[Z-transform]] · Complex frequency · S-plane · Post's inversion formula <!-- VISUAL-LINK:START --> ## From the Real GENERATIVE library > In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually t {\displaystyle t} , in the time domain) to a function of a complex variable s {\displaystyle s} (in the complex-valued frequency domain, also known as s-domain, or s-plane). ([Wikipedia](https://en.wikipedia.org/wiki/Laplace_transform)) <!-- REAL-GENERATIVE-MEDIA:END --> <!-- CRAFT-LINK:START g12 --> *Built to the [[WT!P5_js_Microsim_Master_Class|p5.js Master Class]].* <!-- CRAFT-LINK:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Laplace_transform) : [Wikitube](https://en.wikitube.io/wiki/Laplace_transform) ## Previous hub tags Tree parent: [[Control_theory]]. Legacy hubs: none. --- *Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*