How to use the Laplace Transform Calculator

Definition (first line): This Laplace Transform Calculator numerically computes the unilateral Laplace transform F(s)=∫0∞f(t) e−st dtF(s)=∫0∞f(t)e−stdt for a user-specified time-domain function f(t)f(t) and visualizes F(s)F(s) across a real-s range or at a single real s value.

Introduction

The Laplace transform is a cornerstone technique in engineering, control theory, and differential equations. It converts time-domain functions into the complex frequency domain, turning differential equations into algebraic ones for easier solving. For a practical, on-page numeric tool you can use inside a WordPress site, this calculator performs numerical integration of f(t)e−stf(t)e−st up to a configurable upper limit, and plots F(s)F(s) versus a real s-range so you can quickly inspect behavior and approximate analytic results. Authoritative background on the Laplace transform and its common properties is available from mathematical references such as Wolfram MathWorld and university sources. MathWorld+1

What the calculator does (and what it does not)

This calculator:

Accepts a user expression f(t)f(t) using Math.js syntax (e.g., sin(2*t), exp(-3*t)*t^2).
Numerically integrates f(t)e−stf(t)e−st from 0 to a user-configurable upper limit (default 100) using a composite Simpson numerical integrator for stable numeric results.
Computes either a single numeric value F(s)F(s) for an entered real s, or computes and plots F(s)F(s) across a real s range (start, end, and number of points).
Uses Plotly.js for responsive, interactive plotting embedded in your WordPress page. plotly.com

It does not symbolically compute closed-form Laplace transforms (for that, consult CAS tools or table lookups). The numerical approach is intentionally general-purpose, letting you explore transforms for functions that might be messy to integrate analytically.

Quick start — step-by-step

In the function box, type your time-domain function using t as the variable. Examples:
1 (unit step), sin(2*t), t^2, exp(-3*t) * t.
Math.js expression rules allow standard math functions and operators.
If you want the transform at a single real s, enter the value in the Compute single s value field (e.g., 1.5) and click Compute. The numeric result will appear.
To inspect behavior across a range of s values, leave the single s field empty and fill s start, s end, and points (e.g., start 0.1, end 20, points 200). Click Compute — the chart will render with an interactive hover readout.
If a function decays slowly (or does not decay), increase the upper limit (default 100) and/or increase steps for more accuracy. The integration is truncated at the upper limit, so choose it large enough that the integrand f(t)e−stf(t)e−st is negligible beyond that limit.
Interpret results: the plotted values approximate the real part of the transform for real s. For most engineering use cases and causal functions, real s > 0 is the relevant region.

Practical tips & accuracy

Functions with exponential decay (e.g., exp(-a*t)) converge quickly — smaller upper limits are fine. For slowly decaying functions or pure polynomials multiplied by small exponential decay, you may need larger upper limits and more steps.
Use moderate numbers of integration steps (e.g., 1000–5000) for a balance of speed and accuracy; more steps increase CPU usage because the integration is performed in the browser.
This numeric method computes the unilateral transform (t≥0). For bilateral transforms or full complex s-plane analysis a symbolic CAS or contour methods are required. For conceptual lessons and video tutorials, reputable resources such as Khan Academy provide clear instructional material. Khan Academy

Why Plotly and Math.js?

Plotly.js provides interactive, responsive plotting with zoom, pan, and hover tooltips that make it easy to inspect transform behavior across frequencies; it’s served from a CDN for fast loading. Math.js enables safe parsing and evaluation of user expressions in the browser so you can enter standard mathematical expressions without server-side parsing. Both choices support a good page experience (fast load when CDN cached, mobile-friendly, and secure when served over HTTPS). plotly.com+1

Security & page experience notes

All computations run client-side; no user data is sent to servers. This keeps privacy high and simplifies WordPress integration.
Use HTTPS on your site (WordPress generally provides this) to properly load CDN scripts and keep the page secure.
Keep the number of integration steps and plotting points reasonable to avoid long-running scripts in the browser.

External references (authoritative)

Wolfram MathWorld — Laplace Transform (definition & properties). MathWorld
Wikipedia — Laplace transform (overview & tables). Wikipedia
Khan Academy — Laplace transform lessons (guided explanation & examples). Khan Academy
Plotly.js — Getting started & CDN guidance (plotting library used). plotly.com

FAQ

Q1: Can the calculator handle complex s (e.g., s = a + ib)?
A1: The provided implementation currently evaluates real-valued s only. Extending to complex s requires a complex evaluator and complex arithmetic integration; it’s possible but more advanced. For many practical uses (stability and amplitude analysis), inspecting real s is sufficient.

Q2: Why does the numeric result differ slightly from textbook tables?
A2: Textbook values are exact closed forms; numeric integration is an approximation truncated at the upper limit and with finite step size. Improve match by increasing upper limit and steps.

Q3: My function diverges — what should I do?
A3: If f(t)f(t) grows faster than eαteαt for some large negative real part, the Laplace transform may not exist for the chosen real s. Choose s large enough (positive) to ensure integrand decays or consult analytic theory.

Q4: Can I export the plotted data?
A4: Plotly provides built-in image export and programmatic data export options; further customization can be added to the tool to download numeric arrays as CSV.

Q5: Is this safe for production WordPress sites?
A5: Yes—dependencies are from official CDNs, and computations are client-side. Always load scripts over HTTPS and test on a staging site before wide deployment.

Laplace Transform Calculator