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Select the parameter you wish to calculate and enter the required ones in the respective fields. The tool will use the law of cosine to determine the results instantly.
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Use this Laplace transform calculator to find the Laplace transformation of a function f(t) or an ordinary differential equation (ODE). The calculator applies relevant formulas and integral operations to provide accurate results with detailed steps.
Laplace transform is a mathematical technique that converts a time-domain function f(t) into a function of a complex variable s. It is widely used in physics, engineering, and control theory to solve ODEs.
Mathematically:
F(s) = ∫₀^∞ f(t) e^{-st} dt
Where:
F(s) = ∫₀^∞ f(t) e^{-st} dt
Example:
Given: f(t) = 6e^{-5t} + e^{3t} + 5t³ - 9
Step 1: Apply the Laplace formula
F(s) = ∫₀^∞ (6e^{-5t} + e^{3t} + 5t³ - 9) e^{-st} dt
Step 2: Solve each term individually
Step 3: Combine all terms
F(s) = 6 / (s + 5) + 1 / (s - 3) + 30 / s⁴ + 9 / s
To convert back to the time domain, use the Inverse Laplace Transform Calculator.
Common Laplace transforms:
| Function | Time-domain f(t) | Laplace Transform F(s) |
|---|---|---|
| Constant | 1 | 1/s |
| Linear | t | 1/s² |
| Power | tⁿ | n!/s^(n+1) |
| Exponent | e^(at) | 1/(s-a) |
| Sine | sin(at) | a / (s² + a²) |
| Cosine | cos(at) | s / (s² + a²) |
| Hyperbolic sine | sinh(at) | a / (s² - a²) |
| Hyperbolic cosine | cosh(at) | s / (s² - a²) |
| Growing sine | t sin(at) | 2as / (s² + a²)² |
| Growing cosine | t cos(at) | (s² - a²) / (s² + a²)² |
| Decaying sine | e^(-at) sin(ωt) | ω / ((s+a)² + ω²) |
| Decaying cosine | e^(-at) cos(ωt) | (s+a) / ((s+a)² + ω²) |
| Delta function | δ(t) | 1 |
| Delayed delta | δ(t-a) | e^(-as) |
| Property | Equation |
|---|---|
| Linearity | L{f(t) + g(t)} = F(s) + G(s) |
| Time Delay | L{f(t-td)} = e^(-s td) F(s) |
| First Derivative | L{f'(t)} = s F(s) - f(0-) |
| Second Derivative | L{f''(t)} = s² F(s) - s f(0-) - f'(0-) |
| Nth Derivative | L{f^(n)(t)} = s^n F(s) - s^(n-1) f(0-) - ... - f^(n-1)(0-) |
| Integration | L{∫f(t) dt} = 1/s F(s) |
| Convolution | L{f(t) * g(t)} = F(s) G(s) |
| Initial Value Theorem | lim(s→∞) s F(s) = f(0-) |
| Final Value Theorem | lim(s→0) s F(s) = f(∞) |
Laplace Transform converts a time-domain signal into a complex frequency-domain signal. Fourier Transform converts it into the 'jw' complex plane, a special case of Laplace Transform when the real part is zero.
Yes. If f(t) = 0, then F(s) = 0, following the linearity property of the Laplace Transform.
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