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Free Laplace transform calculator & inverse Laplace transform calculator. Convert time-domain functions to s-domain with step-by-step solutions, convergence analysis, and transform properties. Perfect for solving differential equations and analyzing control systems.
Last updated: February 2, 2026
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Enter constants like 1, 5, or any number
F(s) = L{f(t)}
1/s
Region of Convergence:
s > 0
Converges for s > 0
Step-by-Step Solution:
1. Apply Laplace transform definition: L{f(t)} = ∫₀^∞ f(t)e^(-st) dt
2. For f(t) = 1: L{1} = ∫₀^∞ e^(-st) dt
3. Integrate: [-1/s · e^(-st)]₀^∞
4. Evaluate limits: 0 - (-1/s) = 1/s
5. Result: L{1} = 1/s
Common Laplace Transforms:
L{1} = 1/s
L{t} = 1/s²
L{t^n} = n!/s^(n+1)
L{e^(at)} = 1/(s-a)
L{sin(at)} = a/(s²+a²)
L{cos(at)} = s/(s²+a²)
Laplace Transform Properties:
Learning feature
Detailed Solution Process
See every integration step and property application explained clearly
Inverse operation
F(s) → f(t)
Use transform tables and partial fractions for inverse transforms
Application
ODE → Algebra → Solution
Transform differential equations into solvable algebraic equations
Quick reference
50+ Standard Transforms
Access comprehensive tables of common Laplace transform pairs
Complex analysis
s = σ + jω
Work with complex frequency domain for control system analysis
System analysis
H(s) = Y(s)/X(s)
Derive transfer functions for control systems and signal processing
Laplace transform of constant function f(t) = 1:
Transform
F(s) = 1/s
Convergence
s > 0
L{1} = ∫₀^∞ e^(-st) dt = 1/s for s > 0
Our Laplace transform calculator applies the integral transform definition to convert time-domain functions to the s-domain (complex frequency domain). The calculator recognizes function types, applies appropriate transform formulas, and provides step-by-step derivations with convergence analysis.
L{f(t)} = F(s) = ∫₀^∞ f(t)e^(-st) dtThe Laplace transform is an integral operator that maps a function f(t) from the time domain to a function F(s) in the complex frequency domain, where s = σ + jω. The transform exists for all s in the region of convergence (ROC).
Linearity: L{af(t) + bg(t)} = aF(s) + bG(s)
First Derivative: L{f'(t)} = sF(s) - f(0)
Integration: L{∫₀^t f(τ)dτ} = F(s)/s
Time Shifting: L{f(t-a)u(t-a)} = e^(-as)F(s)
Frequency Shifting: L{e^(at)f(t)} = F(s-a)
Time domain to frequency domain transformation visualization
The Laplace transform is a powerful mathematical tool developed by Pierre-Simon Laplace. It extends the concept of Fourier transforms to handle a broader class of functions, including those with exponential growth. The transform is particularly useful because it converts calculus operations (differentiation and integration) into algebraic operations, making differential equations much easier to solve.
Need help with other transforms? Check out our Fourier transform calculator and convolution calculator.
Get Custom Calculator for Your PlatformResult:
F(s) = 1/(s + 2)²
Region of Convergence: s > -2
Solve: y' + 2y = e^t, y(0) = 1
Use L{y'} = sY(s) - y(0)
L{sin(3t)} = ?
Result: 3/(s² + 9)
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