Find the time-domain function from a standard-form Laplace transform F(s).
Finds the inverse Laplace transform f(t) = L⁻¹{F(s)} for standard forms. Supported inverse transforms: • L⁻¹{n!/s^(n+1)} = tⁿ • L⁻¹{1/(s-a)} = e^(at) • L⁻¹{ω/(s²+ω²)} = sin(ωt) • L⁻¹{s/(s²+ω²)} = cos(ωt) • L⁻¹{1/(s-a)²} = t·e^(at) • L⁻¹{ω/((s-a)²+ω²)} = e^(at)sin(ωt) • L⁻¹{(s-a)/((s-a)²+ω²)} = e^(at)cos(ωt)
Inverse Laplace
f(t) = L⁻¹{F(s)}
Use partial fraction decomposition to break F(s) into the standard forms listed above, then invert each term separately.
A factor e^(-as) in F(s) corresponds to a time delay: L⁻¹{e^(-as)F(s)} = f(t-a)·u(t-a).