Free Laplace Transform

Laplace Transform Calculator

There is a calculator with detailed steps.

Laplace Transform Calculator

Use this Laplace transform calculator to find the Laplace transformation of a function f(t) or an ordinary differential equation (ODE). Our calculator applies relevant Laplace formulas and integral operations to provide accurate results with detailed steps.

What Is Laplace Transform?

Laplace transform is a mathematical technique used to convert 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:

\(F(s) = \int_{0}^{\infty} f(t)e^{-st} dt\)

Where:

f(t) = time-domain function defined for t ≥ 0
s = complex variable (s = a + bi, where a is real and b is imaginary)
\(\int_{0}^{\infty}\) = improper integral over [0, ∞)
F(s) = function in the frequency domain

How To Find Laplace Transform of a Function?

There are two main methods:

1 - Using Laplace Formula

\(F(s) = \int_{0}^{\infty} f(t)e^{-st} dt\)

Example:

Given:

\(f(t) = 6e^{-5t} + e^{3t} + 5t^3 - 9\)

Step 1: Apply the Laplace formula

\(F(s) = \int_{0}^{\infty} (6e^{-5t} + e^{3t} + 5t^3 - 9) e^{-st} dt\)

Step 2: Solve each term individually

6e^{-5t}: \(\int_{0}^{\infty} 6e^{-5t} e^{-st} dt = 6 \int_{0}^{\infty} e^{-(s+5)t} dt = \frac{6}{s+5}\)
e^{3t}: \(\int_{0}^{\infty} e^{3t} e^{-st} dt = \frac{1}{s-3}\)
5t^3: \(\int_{0}^{\infty} 5t^3 e^{-st} dt = \frac{5 \cdot 3!}{s^4} = \frac{30}{s^4}\)
-9: \(\int_{0}^{\infty} -9 e^{-st} dt = \frac{9}{s}\)

Step 3: Combine all terms

\(F(s) = \frac{6}{s+5} + \frac{1}{s-3} + \frac{30}{s^4} + \frac{9}{s}\)

To convert this back to the time domain, use our inverse Laplace transform calculator.

2 - Using Laplace Transform Calculator

Enter the function f(t) in the input field
Click Calculate
Obtain the frequency-domain function F(s)

Laplace Transform Table

This table shows common Laplace transforms:

Function name	Time-domain f(t)	Laplace transform F(s)
Constant	1	1/s
Linear	t	1/s²
Power	t^n	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)

Properties of Laplace Transform

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(∞)

Applications

Convert time-domain signals into frequency-domain signals for control system design
Verify solutions for complicated Laplace transform problems
Simplify partial differential equations in discrete calculus
Derive moment-generating functions in probability and statistics
Analyze the behavior and stability of electrical circuits

FAQs

What is the difference between Fourier and Laplace Transform?

Laplace Transform converts a time-domain signal into a complex frequency-domain signal. Fourier Transform converts it into the 'jw' complex plane, which is a special case of Laplace Transform when the real part is zero.

Can the Laplace Transform equal 0?

Yes. If f(t) = 0, then F(s) = 0. This follows from the linearity property of the Laplace Transform.

References

Wikipedia: Bilateral Laplace Transform, Inverse Laplace Transform.

Paul's Online Notes: Laplace Transforms, Solving IVPs.

Swarthmore College: Laplace Properties - Linearity, Time Delay, Derivatives, Initial and Final Value Theorems.