Laplace Transform Calculator
Compute Laplace transforms of time-domain functions instantly with detailed step-by-step solutions, common function presets, and comprehensive transform pair reference.
What is the Laplace Transform?
The Laplace transform is an integral transform that converts a function of time $f(t)$ into a function of complex frequency $F(s)$. Named after the French mathematician Pierre-Simon Laplace, this transform is fundamental in engineering, physics, and applied mathematics for solving differential equations and analyzing linear time-invariant systems.
The Laplace transform is defined by the integral:
$$F(s) = \mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t) \, dt$$
The transform converts differentiation and integration in the time domain into simple algebraic operations in the s-domain, making it invaluable for solving complex problems.
How to Use the Laplace Transform Calculator
- Enter the function: Type your time-domain function $f(t)$ using the variable
t. Use standard notation likeexp(-2*t)*sin(3*t). - Use presets: Click any preset button to quickly load common functions for testing or learning. Presets include step, ramp, sine, cosine, damped sine, and hyperbolic functions.
- Compute: Click "Compute Laplace Transform" to calculate $F(s)$ symbolically using our lookup table of transform pairs.
- Review results: Examine the resulting $F(s)$ and the step-by-step derivation showing how the transform was computed.
Common Laplace Transform Pairs
| f(t) | F(s) | Description |
|---|---|---|
| $1$ | $\frac{1}{s}$ | Unit step (constant) |
| $t$ | $\frac{1}{s^2}$ | Ramp function |
| $t^n$ | $\frac{n!}{s^{n+1}}$ | Power function |
| $e^{at}$ | $\frac{1}{s-a}$ | Exponential |
| $\sin(bt)$ | $\frac{b}{s^2+b^2}$ | Sine function |
| $\cos(bt)$ | $\frac{s}{s^2+b^2}$ | Cosine function |
| $e^{-at}\sin(bt)$ | $\frac{b}{(s+a)^2+b^2}$ | Damped sine |
| $e^{-at}\cos(bt)$ | $\frac{s+a}{(s+a)^2+b^2}$ | Damped cosine |
| $te^{at}$ | $\frac{1}{(s-a)^2}$ | t times exponential |
| $\sinh(at)$ | $\frac{a}{s^2-a^2}$ | Hyperbolic sine |
| $\cosh(at)$ | $\frac{s}{s^2-a^2}$ | Hyperbolic cosine |
Key Properties of the Laplace Transform
- Linearity: $\mathcal{L}\{af(t) + bg(t)\} = aF(s) + bG(s)$
- First Derivative: $\mathcal{L}\{f'(t)\} = sF(s) - f(0)$
- Second Derivative: $\mathcal{L}\{f''(t)\} = s^2F(s) - sf(0) - f'(0)$
- Integration: $\mathcal{L}\{\int_0^t f(\tau)d\tau\} = \frac{F(s)}{s}$
- Frequency Shifting: $\mathcal{L}\{e^{at}f(t)\} = F(s-a)$
- Time Shifting: $\mathcal{L}\{f(t-a)u(t-a)\} = e^{-as}F(s)$
- Convolution: $\mathcal{L}\{(f * g)(t)\} = F(s) \cdot G(s)$
Frequently Asked Questions
What is the Laplace transform used for?
The Laplace transform is used to convert differential equations into algebraic equations, making them easier to solve. It is widely applied in control systems engineering, electrical circuit analysis, signal processing, heat transfer, and mechanical vibration analysis.
What is the difference between the Laplace transform and the Fourier transform?
The Fourier transform is a special case of the Laplace transform when $s = j\omega$ (purely imaginary). The Laplace transform is more general and can handle functions that grow exponentially, while the Fourier transform requires functions to be absolutely integrable.
What does the region of convergence (ROC) mean?
The region of convergence (ROC) is the set of values of $s$ for which the Laplace transform integral converges. The ROC is essential for determining system stability and for uniquely identifying the original function from its transform. For causal signals, the ROC extends to the right of the rightmost pole.
How do I enter functions in this calculator?
Use t as the time variable. The calculator supports: exp(x) for the exponential function, sin(x), cos(x) for trigonometric functions, sinh(x), cosh(x) for hyperbolic functions, sqrt(x) for square root, and ^ for powers. Use * for multiplication.
Can I compute inverse Laplace transforms with this tool?
This tool computes the forward Laplace transform f(t) to F(s). For the inverse operation (F(s) to f(t)), try the Inverse Laplace Transform Calculator.
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