Laplace Transform Calculator
Compute the Laplace transform of standard forms: polynomials, exponentials, trig and steps.
Inputs
F(s) value
0.25000000
L{f(t)} = F(s)
L{t^2} = 2! / s^3 = 2 / s^3
Step by step
Values used
Function type = tⁿ (power); n (power/order) = 2; a (exponential rate / step delay) = 1; ω (angular frequency) = 1; Evaluate F(s) at s = = 2
Laplace transform definition
F(s) = ∫₀^∞ f(t)·e^(-st) dt
F(s) value
= 0.25000000
L{f(t)} = F(s)
= L{t^2} = 2! / s^3 = 2 / s^3
How it works
Computes the Laplace transform F(s) = ∫₀^∞ f(t)e^(-st)dt for standard function forms. Supported transforms: • L{tⁿ} = n!/s^(n+1) • L{e^(at)} = 1/(s-a) • L{sin(ωt)} = ω/(s²+ω²) • L{cos(ωt)} = s/(s²+ω²) • L{t·e^(at)} = 1/(s-a)² • L{e^(at)sin(ωt)} = ω/((s-a)²+ω²) • L{e^(at)cos(ωt)} = (s-a)/((s-a)²+ω²) • L{u(t-a)} = e^(-as)/s
Formula
Laplace transform definition
F(s) = ∫₀^∞ f(t)·e^(-st) dt
- F(s)
- Transform in s-domain
- f(t)
- Original time-domain function
- s
- Complex frequency variable
Frequently Asked Questions
What are Laplace transforms used for?
They convert differential equations into algebraic equations in the s-domain, making them much easier to solve. Widely used in control systems and signal processing.
What does the convergence condition s > a mean?
The integral only converges when the real part of s exceeds the growth rate a of the exponential. This defines the region of convergence (ROC).