Laplace Transform Calculator

ompute the unilateral Laplace transform for common signals. Visualize poles, zeros, and Region of Convergence (ROC).

Laplace Transform
Inverse Laplace Transform
Properties

Laplace Transform Formula: \[ \mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} f(t)e^{-st} dt \]

Where: f(t) = function of time, s = complex frequency parameter

f(t) =
Enter a function of time t. Use standard math notation: * for multiplication, ^ for exponentiation, sin(), cos(), exp(), etc.
t²
t³
sin(a·t)
cos(ω·t)
e^(-a·t)
t·e^(-t)
sinh(k·t)
cosh(k·t)
1 (constant)
t
e^(-t)·sin(t)
e^(-t)·cos(t)
Variable for the transformed function (usually s)
Variable for the time domain function (usually t)

Inverse Laplace Transform Formula: \[ \mathcal{L}^{-1}\{F(s)\} = f(t) = \frac{1}{2\pi i} \int_{\gamma-i\infty}^{\gamma+i\infty} F(s)e^{st} ds \]

Where: F(s) = function in s-domain, t = time variable

F(s) =
Enter a function of complex frequency s. Use standard math notation.
1/s
1/s²
1/(s² + a²)
s/(s² + a²)
1/(s - a)
1/(s² - a²)
s/(s² - a²)
e^(-a·s)/s
1/(s·(s+1))
(s+2)/((s+1)(s+3))

Laplace Transform Properties Calculator

Apply various Laplace transform properties to functions

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The Laplace Transform: Foundation of Modern Engineering

The Laplace transform converts differential equations into algebraic equations in the complex frequency domain (s-domain). Defined as F(s)=∫0∞f(t)e−stdt, it is indispensable for analyzing linear time-invariant (LTI) systems, control loops, signal processing, and electrical networks. This calculator provides exact symbolic transforms, Region of Convergence (ROC), and visual pole-zero maps — essential for stability analysis and frequency response.

Why ROC matters: The same algebraic expression 1s+1 can represent different time functions (causal vs anti-causal) depending on ROC. Our calculator gives the ROC for causal signals (standard engineering context).

Complete derivation & step-by-step philosophy

For each selected function, the tool applies the definition or standard transform pairs verified by authoritative references (Abramowitz & Stegun, Oppenheim's Signals & Systems). For example, L{e−atu(t)}=1s+a with ROC Re(s)>−a. The pole at s=−a dictates system stability. The pole-zero plot reveals frequency response — poles in left-half plane guarantee stability for causal systems.

Real-world application spotlight

Control Systems: PID tuning & stability
A plant transfer function G(s)=ωn2s2+2ζωns+ωn2 is derived via Laplace transform of differential equations. Its poles determine damping ratio ζ and natural frequency. Using our calculator to visualize pole locations helps engineers design compensators. The ROC confirms BIBO stability when all poles lie in the left half-plane.
Circuit Analysis Example: RC low-pass filter
Consider a simple RC series circuit with output voltage across the capacitor. The transfer function is H(s)=1/RCs+1/RC. Using our calculator with a=1/RC (e.g., R=1kΩ,C=1μF, then a=1000), you obtain H(s)=1000s+1000. The pole at s=−1000 and ROC Re(s)>−1000 confirm causality. The -3 dB cutoff frequency fc=10002π≈159Hz is directly read from the pole magnitude. This example is standard in electronics textbooks (see Sedra & Smith, “Microelectronic Circuits”).

Complete Laplace Transform Table (verification ready)

f(t) (for t≥0)F(s)ROCPoles
δ(t)1All sNone
u(t)1/sRe(s) > 00
tnn!/sn+1Re(s) > 00 (order n+1)
e−at1/(s+a)Re(s) > -a-a
sin⁡(ωt)ω/(s2+ω2)Re(s) > 0±jω
e−atsin⁡(ωt)ω/[(s+a)2+ω2]Re(s) > -a−a±jω

Frequently Asked Questions

Pole-zero cancellation can mask unstable modes. However, in physical systems exact cancellation is impossible due to parameter variations. Our pole-zero plot helps identify hidden dynamics.

For bounded-input bounded-output (BIBO) stability, all poles must lie strictly in the left half-plane. Poles on the imaginary axis (e.g., ±jω) lead to sustained oscillations (marginally stable). This calculator shows ROC accordingly.

This tool focuses on forward Laplace transform. For inverse, you can use partial fraction expansion with the provided pole-zero data. A dedicated inverse Laplace calculator is coming soon.

All calculations use double-precision floating point (IEEE 754). Complex poles are displayed symbolically (e.g., -2 ± j3); the plot maps them exactly to pixel coordinates using linear scaling. The numeric precision is better than 1e-12 relative error, which is far beyond engineering requirements.
Authoritative references & further reading:
– A. V. Oppenheim, A. S. Willsky, S. H. Nawab, Signals and Systems (2nd ed.), Prentice Hall, 1996 (ISBN 0-13-814757-4).
– M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, 1965 (Chapter 29).
– Wolfram MathWorld: Laplace Transform
– Swarthmore College: The Laplace Transform Tutorial