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Laplace Transform Calculator

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Laplace Transform Calculator

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.

What Is Laplace Transform?

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(t) = time-domain function defined for t ≥ 0
s = complex variable (s = a + bi, with a real and b imaginary)
∫₀^∞ = improper integral over [0, ∞)
F(s) = function in the frequency domain

How To Find Laplace Transform of a Function?

1. Using Laplace Formula

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

6e^{-5t}: ∫₀^∞ 6e^{-5t} e^{-st} dt = 6 / (s + 5)
e^{3t}: ∫₀^∞ e^{3t} e^{-st} dt = 1 / (s - 3)
5t³: ∫₀^∞ 5t³ e^{-st} dt = 5·3! / s⁴ = 30 / s⁴
-9: ∫₀^∞ -9 e^{-st} dt = 9 / s

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.

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

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)

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 complex 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, 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, following 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

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