The Inverse Laplace Transform Formula and Its Properties

The inverse Laplace transform formula is the Bromwich integral, a contour integral in the complex plane. Almost no engineer or student evaluates it directly. Instead, decompose F(s) into standard forms and apply a small set of properties: linearity, shifting, and convolution. That recovers f(t) without integrating. Here is the formal definition and the rules you will actually use.

The inverse Laplace transform is defined by the Bromwich integral (Churchill, Operational Mathematics, §6):

ℒ⁻¹{F(s)} = f(t) = (1 / (2πi)) ∫_{c-i∞}^{c+i∞} F(s) e^{st} ds for t ≥ 0

The integration runs along a vertical line Re(s) = c, where c lies to the right of all singularities of F(s). Use the Bromwich integral only when F(s) is non‑rational, for example, when it contains terms like e^{−as} or log(s). For rational F(s), the residue method (Churchill §7) converts the contour integral into a sum of residues at the poles of F(s). In practice, that is identical to performing a partial‑fraction decomposition and looking up the result.

The four properties below, plus the convolution theorem, are the tools that let you invert rational F(s) without writing a contour integral. Each property is demonstrated with a one‑line example.

Linearity

ℒ⁻¹{αF(s) + βG(s)} = α f(t) + β g(t).

First Shifting Theorem

ℒ⁻¹{F(s−a)} = e^{at} f(t).

The shift in s multiplies the time‑domain result by e^{at}.

Second Shifting Theorem

ℒ⁻¹{e^{−as}F(s)} = u(t−a) f(t−a), where u(t−a) is the unit step function.

Example: ℒ⁻¹{e^{−2s}·1/(s+3)} = u(t−2) e^{-3(t−2)}.

Derivative in s

ℒ⁻¹{F'(s)} = −t f(t).

The convolution theorem states ℒ⁻¹{F(s)G(s)} = (f * g)(t) = ∫₀ᵗ f(τ)g(t−τ) dτ. It is the fundamental reason the s‑domain product corresponds to a time‑domain convolution. For a full worked example, refer to the treatment of the convolution theorem in Boyce & DiPrima §6.6. For most practical inversions, use the residue method instead: the inverse of a rational F(s) is the sum of the residues of F(s)e^{st} at its poles. That is what partial‑fraction decomposition is doing algebraically. Churchill §7 gives the formal statement.