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Laplace Transform Calculator
Last updated: May 31, 2026
Written by Blake Boege
A Laplace transform calculator is a mathematical tool that converts functions of a real variable (typically time, t) into a complex frequency-domain representation (s). It evaluates functions using a standard lookup table of transform pairs combined with linearity and shift theorem properties. This integral transform simplifies system analysis by turning differential equations into linear algebraic equations. Engineers and physicists use it to analyze electrical circuits, control systems, and transient state responses in physical systems.
Compute the Laplace transform F(s) of common functions of time f(t). View step-by-step shifts and formulas using standard transform pairs.
Quick Answer
Compute the Laplace transform of standard time-domain functions. Enter your function f(t) to find its s-domain transform F(s) using transform pairs.
Supports common functions in variable t.
Try Examples
Step-by-step Explanation
F(s) = L{f(t)} =
2/(s² + 4)
Formula: L{sin(at)} = a / (s² + a²)
Notice: This calculator uses standard pairs and linearity for common functions. For general symbolic Laplace transforms, refer to full CAS software.
Examples
sin(2t)
2 / (s² + 4)
e^(3t)
1 / (s - 3)
t²
2 / s³
cos(5t)
s / (s² + 25)
e^(-2t) * sin(3t)
3 / ((s + 2)² + 9)
How it works
The calculator normalizes the function $f(t)$ and checks it against a lookup table of standard Laplace transform pairs. It also applies the linearity property to handle sums of terms.
Integral Definition
F(s) = L{f(t)} = ∫₀^∞ e⁻ˢᵗ f(t) dt
Common Pairs
- L{1} = 1/s
- L{t} = 1/s²
- L{e^(at)} = 1/(s - a)
- L{sin(at)} = a/(s² + a²)
- L{cos(at)} = s/(s² + a²)
Applications in Control Systems & Engineering
In electrical engineering and control systems, the Laplace transform is used to analyze transfer functions of circuits and mechanical models. It helps determine stability, transient response, and frequency response without solving complex differential equations in the time domain.
Common Limitations
Not all functions have a Laplace transform. The integral definition requires that f(t) does not grow faster than exponentials as t approaches infinity (it must be of exponential order). For functions that grow extremely fast, like e^(t²), the Laplace integral does not converge.
Related Mathematical Tools
If you are working on numerical double integration, visit our Double Integral Calculator. For derivatives or rate of change analyses, check out our Rate of Change Calculator.
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Frequently asked questions
The Laplace transform is an integral transform that converts a function of a real variable t (often representing time) into a function of a complex variable s (representing complex frequency).
The Laplace transform of f(t) is defined by the improper integral: F(s) = ∫₀^∞ e^(-st) f(t) dt, for s > 0.
It is widely used to solve linear differential equations. By converting differential operations in the time domain into simple algebraic operations in the s-domain, it simplifies system analysis in engineering and physics.
The Linearity Property states that the Laplace transform of a sum of functions scaled by constants is equal to the sum of their individual Laplace transforms: L{a*f(t) + b*g(t)} = a*L{f(t)} + b*L{g(t)}.
The First Shift Theorem states that multiplying a function by an exponential e^(at) shifts its Laplace transform in the s-domain: L{e^(at) f(t)} = F(s - a), where F(s) is the Laplace transform of f(t).
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