๐ Laplace Transform Calculator
Calculate forward and inverse Laplace transforms instantly. Perfect for differential equations, control systems, and engineering mathematics.
Input Function f(t)
Common Laplace Transforms
| Name | f(t) | โ{f(t)} = F(s) | Conditions | |
|---|---|---|---|---|
| Constant | c | c/s | s > 0 | |
| Exponential | e^(at) | 1/(s-a) | s > a | |
| Power | t^n | n!/(s^(n+1)) | s > 0, n โ โ | |
| Sine | sin(at) | a/(sยฒ+aยฒ) | s > 0 | |
| Cosine | cos(at) | s/(sยฒ+aยฒ) | s > 0 | |
| Sinh | sinh(at) | a/(sยฒ-aยฒ) | s > |a| | |
| Cosh | cosh(at) | s/(sยฒ-aยฒ) | s > |a| | |
| Unit Step | u(t-a) | e^(-as)/s | s > 0 | |
| Delta Function | ฮด(t-a) | e^(-as) | all s | |
| t*exp | t*e^(at) | 1/(s-a)ยฒ | s > a |
Understanding Laplace Transforms
What is a 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). It is defined as:
โ{f(t)} = F(s) = โซโ^โ f(t)e^(-st) dt
where s is a complex number (s = ฯ + jฯ)
How to Calculate Laplace Transform
To calculate the Laplace transform of a function:
- Identify the function type - Check if it matches a standard form in the transform table
- Apply linearity - Break complex functions into simpler parts using โ{af + bg} = aF(s) + bG(s)
- Use properties - Apply shifting theorems, derivative/integral properties as needed
- Consult the table - Look up each component in the Laplace transform table
- Combine results - Add or multiply transforms according to properties used
How to Calculate Inverse Laplace Transform
Finding the inverse Laplace transform involves working backwards from F(s) to f(t):
- Simplify F(s) - Factor denominators and look for recognizable patterns
- Use partial fractions - Decompose complex rational functions into simpler terms
- Complete the square - Rewrite quadratic expressions to match standard forms
- Match to table - Identify each term with a known inverse transform
- Apply inverse properties - Use shifting theorems in reverse
- Combine results - Sum the individual inverse transforms using linearity
Laplace Transform of Piecewise Functions
Piecewise functions require special treatment using unit step functions:
Example: Piecewise Function
f(t) = {0, t < a; g(t), t โฅ a}
Step 1: Rewrite using unit step: f(t) = g(t)u(t-a)
Step 2: Express g(t) in terms of (t-a): g(t) = h(t-a) + constant
Step 3: Apply second shifting theorem: โ{f(t)} = e^(-as)H(s)
Applications of Laplace Transforms
- Differential Equations: Convert ODEs to algebraic equations in s-domain
- Control Systems: Analyze system stability and transfer functions
- Circuit Analysis: Solve electrical circuits with initial conditions
- Signal Processing: Analyze and filter signals in frequency domain
- Mechanical Systems: Study vibrations and dynamic responses
- Heat Transfer: Solve heat conduction problems with boundary conditions
Step-by-Step Example
Find โ{tยฒe^(3t)sin(2t)}
Step 1: Start with โ{sin(2t)} = 2/(sยฒ+4)
Step 2: Apply first shifting for e^(3t): Replace s with (s-3)
Result: 2/((s-3)ยฒ+4)
Step 3: Use s-domain derivative for tยฒ: Apply (-1)ยฒdยฒ/dsยฒ twice
Final: Complex expression involving derivatives of 2/((s-3)ยฒ+4)
Common Mistakes to Avoid
- Forgetting initial conditions when transforming derivatives
- Incorrect partial fraction decomposition
- Not completing the square for quadratic denominators
- Misapplying shifting theorems (confusing first and second shift)
- Ignoring regions of convergence (ROC)
- Incorrectly handling unit step functions in piecewise functions
Frequently Asked Questions
How to calculate inverse Laplace transform?
To calculate the inverse Laplace transform: 1) Identify the form of F(s), 2) Use partial fraction decomposition if needed, 3) Match each term to a known inverse transform from the table, 4) Apply linearity property to combine results. Common techniques include completing the square, partial fractions, and using the shift theorem.
How to calculate Laplace transform?
Calculate the Laplace transform using the integral definition: โ{f(t)} = โซโ^โ f(t)e^(-st)dt. For common functions, use the transform table to directly find F(s). Apply linearity for sums, use the first and second shifting theorems for shifted functions, and the derivative/integral properties for derivatives and integrals of f(t).
How to calculate Laplace transform of piecewise function?
For piecewise functions: 1) Express each piece using unit step functions u(t-a), 2) Rewrite the piecewise function as a sum of shifted functions, 3) Apply Laplace transform to each term using the second shifting theorem: โ{f(t-a)u(t-a)} = e^(-as)F(s), 4) Combine all terms to get the final transform.
What is the Laplace transform used for?
Laplace transforms are used to solve differential equations, analyze linear time-invariant systems, study control systems, solve initial value problems, and convert convolution to multiplication. They transform differential equations from the time domain to the algebraic s-domain, making them easier to solve, then transform back to get time-domain solutions.
Related Math Calculators
Master Differential Equations & Control Systems
Free Laplace transform calculator for students, engineers, and mathematicians
Explore More Calculators