Laplace Transform Calculator
Find Laplace transforms and inverse transforms for common functions.
Direction
Transform Pair
f(t) = e^(at)
F(s) = F(s) = 1 / (s − a)
What Is the Laplace Transform Calculator?
This calculator provides a comprehensive Laplace transform lookup tool with both forward transforms L{f(t)} → F(s) and inverse transforms L⁻¹{F(s)} → f(t). Select a function type, enter parameters, and get the exact symbolic transform pair with conditions of validity.
- ›15 transform pairs: Unit step, delta function, tⁿ, e^(at), sin(bt), cos(bt), sinh(bt), cosh(bt), damped sine/cosine, and more, covering all standard engineering functions.
- ›Bidirectional: Switch between forward (time → frequency domain) and inverse (frequency → time domain) in one click.
- ›Parametric: Enter values for a, b, n and the tool substitutes them into the formula, showing the exact rendered transform pair.
- ›Quick Reference Table: A collapsible table of all 15 transform pairs for fast lookup during problem solving.
- ›Key theorems: Linearity, first-shift, derivative, and integration theorems shown alongside each result for context.
- ›Persistent state: Your last-used function and parameters are saved in browser storage and restored on your next visit.
Formula
Definition (Unilateral Laplace Transform)
F(s) = L{f(t)} = ∫₀^∞ e^(−st) f(t) dt
Key Transform Pairs
L{1} = 1/s (s > 0)
L{tⁿ} = n! / s^(n+1) (n = 0, 1, 2, …)
L{e^(at)} = 1/(s − a) (s > a)
L{sin(bt)} = b/(s² + b²) (s > 0)
L{cos(bt)} = s/(s² + b²) (s > 0)
| Symbol | Name | Description |
|---|---|---|
| f(t) | Time-domain function | The original function defined for t ≥ 0 |
| F(s) | Frequency-domain function | The Laplace transform of f(t), a function of complex variable s |
| s | Complex frequency | Complex number s = σ + jω; for convergence, Re(s) must exceed a threshold |
| e^(−st) | Kernel | The exponential weighting function that damps the integral for large t |
| a | Damping coefficient | Determines decay rate in e^(at) or e^(at)sin(bt) functions |
| b | Frequency | Angular frequency in sin(bt), cos(bt), sinh(bt), cosh(bt) |
How to Use
- 1Choose direction: Select "Forward" to compute L{f(t)}, or "Inverse" to find L⁻¹{F(s)}.
- 2Select a function: Pick from 14 function types in the dropdown, e^(at), sin(bt), tⁿ·e^(at), damped oscillations, and more.
- 3Enter parameters: Type values for a (damping), b (frequency), or n (power) as they appear. Defaults are pre-filled.
- 4Press Enter or click Calculate: The transform pair appears with your specific parameter values substituted in.
- 5Check the conditions: Note the convergence condition (e.g., "s > a") shown with each result, this tells you when F(s) is valid.
- 6Open the reference table: Click "Quick Reference Table" to browse all 15 pairs at once, useful when scanning for the form you need.
- 7Copy and reset: Copy the result to clipboard or Reset to clear inputs and saved state.
Example Calculation
Find the Laplace transform of f(t) = e^(−2t) sin(3t)
Function: e^(at) · sin(bt) with a = −2, b = 3
Apply the damped-sine transform pair:
L{e^(at) sin(bt)} = b / [(s−a)² + b²]
Substitute a = −2, b = 3:
L{e^(−2t) sin(3t)} = 3 / [(s − (−2))² + 9]
F(s) = 3 / [(s + 2)² + 9]
Valid for s > −2 (Re(s) must exceed the damping coefficient a)
The First Shifting Theorem
The key insight above comes from the first shifting theorem: L{e^(at) f(t)} = F(s−a). Multiplying f(t) by e^(at) in the time domain is equivalent to shifting F(s) to F(s−a) in the frequency domain. This makes solving damped oscillation ODEs straightforward.
Understanding Laplace Transform
What Is the Laplace Transform?
The Laplace transform converts a function of time f(t) into a function of a complex variable F(s). The transform is defined as F(s) = ∫₀^∞ e^(−st) f(t) dt. The exponential e^(−st) damps the integrand, making the integral converge for a wide class of functions as long as Re(s) is large enough.
The key power of the Laplace transform is that it converts differential equations in the time domain into algebraic equations in the s-domain. Derivatives become multiplications by s, integrals become divisions by s. Once you solve the algebraic equation for F(s), you convert back using the inverse transform.
Why Is It Used in Engineering?
- ›Control systems: Transfer functions H(s) = Y(s)/X(s) characterise how a system responds to inputs. Poles and zeros of H(s) determine stability and transient behaviour.
- ›Circuit analysis: Impedances become Z(s): resistors stay R, capacitors become 1/(sC), inductors become sL. Complex circuits can be solved with Ohm's law in the s-domain.
- ›Signal processing: The s-domain reveals frequency content and filter behaviour. The Fourier transform is a special case of the Laplace transform evaluated on the imaginary axis (s = jω).
- ›ODE solutions: Initial conditions are automatically incorporated into the transform, eliminating the need for separate particular and homogeneous solutions.
Key Properties and Theorems
- ›Linearity: L{af(t) + bg(t)} = aF(s) + bG(s), transforms can be combined for sums of functions.
- ›First shifting theorem: L{e^(at)f(t)} = F(s−a), multiplication by an exponential shifts F(s).
- ›Derivative theorem: L{f′(t)} = sF(s) − f(0), each derivative introduces a factor of s.
- ›Convolution: L{f * g} = F(s)·G(s), convolution in time equals multiplication in the s-domain.
- ›Final value theorem: lim(t→∞) f(t) = lim(s→0) sF(s), useful for finding steady-state values without inverting.
Frequently Asked Questions
What is the Laplace transform used for?
The Laplace transform is used primarily to solve linear ordinary differential equations with initial conditions. It converts a differential equation into an algebraic equation in the complex s-domain, which is much easier to solve.
It is essential in electrical engineering (circuit analysis), control engineering (transfer functions), mechanical engineering (vibration analysis), and signal processing (filter design).
What is the difference between Laplace and Fourier transforms?
The Fourier transform F(ω) = ∫₋∞^∞ f(t) e^(−jωt) dt uses a purely imaginary exponent (jω) and integrates over all time.
The Laplace transform uses s = σ + jω (a complex exponent) and integrates only for t ≥ 0. The Fourier transform is the Laplace transform evaluated on the imaginary axis (σ = 0). Laplace handles growing functions and initial conditions more naturally.
What does "region of convergence" mean?
The region of convergence (ROC) is the set of complex s values for which the Laplace integral converges. For causal exponential functions, the ROC is typically a right half-plane: Re(s) > a, where a is the exponential growth rate.
This calculator shows the condition (e.g., "s > 2") for each transform pair. You need s to satisfy this condition for F(s) to be a valid, finite result.
How do I find the Laplace transform of a derivative?
The derivative theorem states: L{f'(t)} = sF(s) − f(0). For the second derivative: L{f''(t)} = s²F(s) − sf(0) − f′(0).
This is why Laplace transforms are so useful for ODEs, each differentiation becomes multiplication by s, and the initial conditions appear automatically as constants in the algebraic equation.
What is the inverse Laplace transform?
The inverse Laplace transform recovers f(t) from F(s). The formal definition uses a complex contour integral (Bromwich integral), but in practice engineers use partial fraction decomposition plus transform tables.
For example, F(s) = 1/(s−a)(s−b) is decomposed into A/(s−a) + B/(s−b), each term matched to a table entry giving f(t) = Ae^(at) + Be^(bt).
What is a transfer function?
A transfer function H(s) = Y(s)/X(s) is the ratio of the output to input Laplace transforms, assuming zero initial conditions. It fully characterises a linear time-invariant (LTI) system.
Poles of H(s) (values where H(s) → ∞) determine stability: if all poles have negative real parts, the system is stable. Zeros affect the frequency response shape.
Does this calculator do symbolic integration?
No, this tool provides transform pairs from standard tables. You select a function type (e^(at), sin(bt), etc.) and enter parameters, and the calculator looks up and renders the corresponding symbolic Laplace transform pair with your values substituted in.
This covers the vast majority of functions encountered in engineering courses and practice. For unusual composite functions requiring integration from first principles, consult a computer algebra system (CAS) such as Wolfram Alpha or MATLAB Symbolic Toolbox.