Inverse Laplace Transform Calculator

The common assumption is that finding an inverse Laplace transform means flipping through a table until an exact match appears. In engineering and differential equations work, the inverse Laplace transform calculator does the heavy lifting by decomposing a given F(s) into standard forms using partial fractions and the shifting theorems, then applying the table. The calculator recovers f(t) from F(s) for t ≥ 0, showing every algebraic step. It handles basic transforms, rational functions with denominators up to degree 2, shifted and exponentially multiplied forms, and derivative-in-s-domain cases. The result includes the time-domain expression and an optional plot over a selected range.

The inverse Laplace transform takes a function F(s) in the complex frequency domain (the s-domain) and returns the original function f(t) in the time domain. Notation: ℒ⁻¹{F(s)} = f(t), defined for t ≥ 0. This is the operation that lets you convert the algebraic solution of a differential equation back into the time-dependent behaviour you actually need, the position of a mass on a spring, the voltage across a capacitor, the output of a control system.

The definition itself is the Bromwich integral: f(t) = (1/(2πi)) ∫_{c−i∞}^{c+i∞} F(s) e^{st} ds, for c greater than the abscissa of convergence. In practice, engineers and students never compute that integral. They use a combination of three methods that the calculator automates: table lookup, partial-fraction decomposition, and the shifting theorems. The result is always accompanied by the step-by-step working that a textbook like Boyce & DiPrima 'Elementary Differential Equations' or Kreyszig 'Advanced Engineering Mathematics' would require.

Why t ≥ 0

The Laplace transform integrates from 0 to ∞, so the inverse is naturally defined only for non‑negative time. Negative t is not considered. The calculator multiplies every inverse by the unit step function u(t) implicitly, making f(t) = 0 for t < 0. This convention matches every standard table and every engineering application.

Selecting the right transform type is the only real decision you make. The calculator provides six modes, each designed for a specific class of F(s).

Basic Transforms

Choose this when your F(s) matches one of the ten pairs exactly: 1/s, n!/s^(n+1), 1/(s−a), ω/(s²+ω²), s/(s²+ω²), ω/(s²−ω²), s/(s²−ω²), ω/((s−a)²+ω²), or (s−a)/((s−a)²+ω²). Enter the parameters a, ω, and an optional multiplier A. The calculator returns the corresponding f(t) directly with no algebra shown, the table entry is the working.

Rational Function (Partial Fractions)

This is the mode you will use most for differential equations. Enter the numerator and denominator coefficients as quadratics. The calculator does polynomial division when the numerator degree is ≥ the denominator degree, producing δ(t) terms (impulses) from the quotient. It then decomposes the proper remainder using partial fractions for denominators of degree 1 or 2, distinct real roots, repeated real roots, or complex‑conjugate roots. The step‑by‑step solution shows the discriminant, the cover‑up coefficients, and each inversion. If the denominator expands to degree 3 or higher, the calculator rejects it: you must split the factors manually and use Custom Expression.

Shifted Function (First Translation Theorem)

Use this when F(s) is of the form G(s−a), the frequency shift theorem. Pick a base function G(s) from the list, enter the shift a, and the calculator returns e^(at)·g(t). This is the route for transforms that look like 1/(s−a), ω/((s−a)²+ω²), and so on.

Exponential Multiplied (Second Translation Theorem)

For delays, F(s) = e^(−as)·G(s). Enter the delay a and the base G(s). The calculator applies ℒ⁻¹{e^(−as)G(s)} = u(t−a)·g(t−a). The result includes the unit step function explicitly.

Derivative in s‑Domain

When F(s) is the first or second derivative of a known transform. Pick the original F(s) and the derivative order. The calculator uses ℒ⁻¹{F'(s)} = −t·f(t) and ℒ⁻¹{F''(s)} = t²·f(t).

Custom Expression

For rational functions that are already split into a sum of partial fractions, e.g. 1/(s+1) + 2/(s+3). The expression parser accepts numbers, s, + − * / ^ and parentheses. The denominator of each term after expansion must be degree 2 or less. Exponentials like e^(−2s) are not parsed here, use the Second Translation mode for delays.

The calculator uses three techniques, applied in a fixed order. The step‑by‑step output labels which method is used at each stage.

Table Lookup

The ten pairs listed in the Basic Transforms mode come directly from the tables in Boyce & DiPrima (11th ed., Table 6.1.1) and Kreyszig (10th ed., Table 6.9).The eight most‑used pairs appear in the table section below.

Partial Fractions

When the table does not contain the exact F(s), the calculator decomposes it. The decomposition follows the forms from Stewart's 'Calculus' §7.4: distinct linear factors, repeated linear factors, irreducible quadratics. The Heaviside cover‑up method (Boyce & DiPrima §6.3) finds residues for distinct real poles. For complex‑conjugate poles, the calculator completes the square and rewrites the numerator as a combination of (s−a) and a constant times ω, producing damped sine and cosine terms.

Shifting Theorems

The first translation (frequency shift) theorem, ℒ⁻¹{F(s−a)} = e^(at)f(t), and the second translation (time shift) theorem, ℒ⁻¹{e^(−as)F(s)} = u(t−a)f(t−a), are applied when the user selects the corresponding mode. Both are used internally in the Rational Function mode when the transform contains factors like (s−a) or e^(−as).

Consider F(s) = (s+3)/(s²+2s+5). The numerator degree (1) is less than the denominator degree (2), so no polynomial division is needed. The discriminant of the denominator is 2² − 4·5 = 4 − 20 = −16, so the roots are complex conjugates: s = −1 ± 2i. The calculator completes the square: s²+2s+5 = (s+1)² + 2². It rewrites the numerator as (s+1) + 2, so the fraction becomes (s+1)/((s+1)²+2²) + 2/((s+1)²+2²). The second term is 2/((s+1)²+2²) = 1·2/((s+1)²+2²), matching the damped sine pair with ω = 2. The first term matches the damped cosine pair.The step‑by‑step output shows the completed square, the discriminant value, the decomposition into the two standard forms, and the final expression.

If the numerator had been degree 2 or higher, say F(s) = (s²+4s+5)/(s²+2s+5), the calculator would first divide, giving a quotient of 1 and a remainder of (2s+0)/(s²+2s+5). The quotient 1 inverts to δ(t), the unit impulse. The remainder then undergoes the same partial‑fraction process.

The inverse Laplace transform is the final step in three workflows: solving ordinary differential equations, analysing electrical circuits, and designing control systems.

Ordinary Differential Equations

Take the Laplace transform of the ODE (including initial conditions), solve for F(s) algebraically, then apply the inverse to get the solution. This turns a differential equation into an algebraic one. The calculator handles the last step, which is the most error‑prone when done by hand. Boyce & DiPrima's entire chapter 6 is built on this method, with exercise sets covering every case type from distinct real roots to convolution.

Electrical Circuits

In circuit analysis, the Laplace transform converts integro‑differential equations (using Kirchhoff's laws) into algebraic equations in s. The inverse recovers the time‑domain current or voltage. For a series RLC circuit with a step input, the inverse gives the transient and steady‑state response. The calculator's Second Translation mode is directly useful when the input is a delayed step or pulse.

Control Systems

The transfer function of a system, the ratio of output transform to input transform with zero initial conditions, is the tool in control engineering. The inverse of the transfer function multiplied by the input transform gives the time‑domain response. The second‑order system form ω_n²/(s²+2ζω_n s+ω_n²) appears constantly; its inverse (for underdamped, critically damped, and overdamped cases) is a core result in Ogata and Nise. The calculator's Rational Function mode handles these directly, showing the damping ratio and natural frequency in the step‑by‑step when the denominator is quadratic.

The single thing that most often goes wrong: a denominator that expands to a cubic or higher is rejected outright. Students pour a fourth‑order polynomial into the Rational Function mode and get an error. The fix is to factor the denominator into quadratics and linear factors yourself, then enter each term separately as a sum in Custom Expression. If the factors are not obvious, use a symbolic factorisation tool first, this calculator is not a factoriser. For any F(s) that requires the Bromwich integral, the residue method, or the Meijer G‑function, switch to a dedicated CAS like Wolfram's InverseLaplaceTransform or SymPy's inverse_laplace_transform. This calculator was built for the cases that appear in differential equations and control textbooks, and it does those well. Outside that set, it stops.