Compute the Laplace transform of standard forms: polynomials, exponentials, trig and steps.
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
Laplace transform definition
F(s) = ∫₀^∞ f(t)·e^(-st) dt
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.
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).