Inverse Laplace Transform Calculator
Find inverse Laplace transforms using standard tables: L⁻¹{1/s}=u(t), L⁻¹{1/(s−a)}=e^(at), L⁻¹{s/(s²+ω²)}=cos(ωt). Includes partial fraction decomposition, final/initial value theorems.
f(t) = L⁻¹{F(s)}
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f(t) at given t —
Extended More scenarios, charts & detailed breakdown ▾
f(t)
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f(t) value —
Professional Full parameters & maximum detail ▾
Partial Fraction Result
f(t) —
f(t) value —
Value Theorems
Final Value Theorem —
Initial Value f(0) —
Applications
Circuit Analogy —
How to Use This Calculator
- Select F(s) from the dropdown (1/s, 1/(s−a), s/(s²+ω²), etc.).
- Enter parameters a and ω.
- Enter a t value to evaluate f(t) numerically.
- Use the Partial Fractions tab for Ae^(p₁t)+Be^(p₂t) form.
- The Professional tab shows final value and initial value theorems.
Formula
L⁻¹{1/(s−a)} = e^(at) | L⁻¹{ω/(s²+ω²)} = sin(ωt) | L⁻¹{s/(s²+ω²)} = cos(ωt)
Example
F(s) = 2/(s+1) + 3/(s+3): f(t) = 2e^(−t) + 3e^(−3t). At t=1: f(1) = 2e⁻¹+3e⁻³ ≈ 0.736+0.149 = 0.885.
Frequently Asked Questions
- Use a transform table to match F(s) to a known form, apply partial fractions to decompose complex fractions, then invert each term. For example: L⁻¹{1/(s−2)} = e^(2t).
- Decompose F(s) = A/(s−p₁) + B/(s−p₂) + ... then L⁻¹{F(s)} = Ae^(p₁t) + Be^(p₂t) + ... Each pole gives an exponential term.
- lim(t→∞) f(t) = lim(s→0) sF(s), provided all poles of sF(s) are in the left half-plane. Used to find steady-state values without inverting.
- f(0⁺) = lim(s→∞) sF(s). Gives the initial value of f(t) directly from F(s) without computing the full inverse.
- The formal inverse is f(t) = (1/2πj)∫_{c−j∞}^{c+j∞} F(s)e^(st)ds. In practice this is evaluated using residue theorem at the poles of F(s).
Related Calculators
Sources & References (5) ▾
- Inverse Laplace — MIT OCW 18.03 — MIT OpenCourseWare
- Schaum's Outline of Laplace Transforms — McGraw-Hill
- NIST DLMF — Laplace Transforms — NIST
- Wolfram MathWorld — Inverse Laplace Transform — Wolfram MathWorld
- Boyce & DiPrima — ODEs Ch. 6 — Wiley