Laplace Transform Calculator
Instantly compute the Laplace transform of common time-domain functions. Convert differential equations into the s-domain for control systems, circuit analysis, and signal processing.
Function Definition
Select a time-domain function f(t) and define parameters
Choose the mathematical form of your time-domain signal
s-Domain Result
Laplace transform F(s) and properties
Select a function and define its parameters, then click Compute Transform to see the s-domain equivalent.
Common Laplace Transform Pairs
Standard time-domain to frequency-domain mappings used in engineering mathematics.
| Time Domain f(t) | s-Domain F(s) | Region of Convergence |
|---|---|---|
| Unit Impulse δ(t) | 1 | All s |
| Unit Step u(t) | 1 / s | Re(s) > 0 |
| tⁿ (n = integer) | n! / sⁿ⁺¹ | Re(s) > 0 |
| eᵃᵗ | 1 / (s – a) | Re(s) > a |
| sin(at) | a / (s² + a²) | Re(s) > 0 |
| cos(at) | s / (s² + a²) | Re(s) > 0 |
| t·eᵃᵗ | 1 / (s – a)² | Re(s) > a |
| eᵃᵗ·sin(bt) | b / ((s – a)² + b²) | Re(s) > a |
| eᵃᵗ·cos(bt) | (s – a) / ((s – a)² + b²) | Re(s) > a |
Laplace Transform FAQ
Everything you need to know about integral transforms, s-domain analysis, and solving differential equations.
The Laplace Transform is a powerful integral transform used to convert a function of time (f(t)) into a function of complex frequency (F(s)). It is defined by the integral from 0 to infinity of e-st * f(t) dt. It is widely used in engineering and physics to solve linear differential equations and analyze control systems.
The primary advantage of the Laplace Transform is that it converts difficult calculus operations (like differentiation and integration) into simple algebraic operations (multiplication and division by ‘s’). This makes solving linear ordinary differential equations much easier, especially for circuits, mechanical vibrations, and control theory.
The Region of Convergence (ROC) is the set of values for the complex variable ‘s’ for which the Laplace Transform integral converges (evaluates to a finite value). For example, the transform of eat is 1/(s-a), which only converges if the real part of s is strictly greater than ‘a’ (Re(s) > a).
The Laplace Transform of the first derivative f'(t) is s*F(s) – f(0). For the second derivative f”(t), it is s²*F(s) – s*f(0) – f'(0). This property is what allows differential equations to be transformed into algebraic equations that can be solved for F(s).
The Fourier Transform is essentially a special case of the Laplace Transform. If you evaluate the Laplace Transform F(s) along the imaginary axis (where s = jω, meaning the real part of s is zero), you get the Fourier Transform. The Laplace Transform is more general because it includes an exponential decay factor that allows it to converge for signals that don’t have a standard Fourier Transform.