How should polynomial coefficients be entered?

Enter coefficients separated by commas in ascending order, beginning with the constant term. For example, enter '2,3,1' to represent s² + 3s + 2.

Does the calculator handle improper rational functions?

No, the function must be a proper rational function where the degree of the numerator is strictly less than the degree of the denominator.

What is the maximum supported denominator degree?

The calculator supports denominator polynomials up to degree 6.

How are complex conjugate roots inverted?

Complex root pairs are retained as irreducible quadratic factors and converted into damped or undamped sine and cosine terms.

Can this tool solve expressions with repeated roots?

Yes, repeated roots are factored into ascending power denominators and inverted using polynomial-exponential pairs.

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Inverse Laplace Transform Calculator (Partial Fractions)

Partial-fraction inverse Laplace transform for proper rational functions: root factorization (real + conjugate pairs), exact coefficient system, term-by-term inversion.

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2 examples

L⁻¹{1/(s² + 3s + 2)} = e^(−t) − e^(−2t)

The denominator factors as (s + 1)(s + 2), so 1/(s² + 3s + 2) = 1/(s + 1) − 1/(s + 2) and each term inverts directly to an exponential: f(t) = e^(−t) − e^(−2t).

Inverse Laplace transform via partial fractions F(s) = 1/(s^2 + 3 s + 2) Step 1 — Denominator roots: r = -1 (×1), r = -2 (×1) Step 2 — Partial fractions: F(s) = 1/(s + 1) - 1/(s + 2) Step 3 — Invert term by term: A/(s - r) → A·e^(rt) f(t) = e^(-t) - e^(-2t)

L⁻¹{(3s + 5)/(s² + 4)} = 3cos(2t) + 2.5sin(2t)

The denominator s² + 4 has the conjugate roots ±2i, so the decomposition stays in quadratic form (3s + 5)/(s² + 4). Shifting to ((s)² + 2²) and inverting the pair gives f(t) = 3·cos(2t) + (5/2)·sin(2t).

Inverse Laplace transform via partial fractions F(s) = (3 s + 5)/(s^2 + 4) Step 1 — Denominator roots: conjugate pair α = 0, β = 2 Step 2 — Partial fractions: F(s) = (3 s + 5)/(s^2 + 4) Step 3 — Invert the quadratic pair: e^(αt)·[B·cos(βt) + ((C + Bα)/β)·sin(βt)] f(t) = 3 cos(2t) + 2.5 sin(2t)

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3 options

ContentPaste or type the main input values.2

Numerator Coefficients N(s) (constant first)textRequiredComma-separated numerator coefficients, constant term first, e.g. 5,3 → 3s + 5.Denominator Coefficients D(s) (constant first)textRequiredComma-separated denominator coefficients, constant term first, degree ≤ 6, e.g. 2,3,1 → s² + 3s + 2.

SettingsAdjust formats, ranges, numbers, and modes.1

Decimal PlacesnumberOptional

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Inverse Laplace Transform Calculator (Partial Fractions)

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Examples

1. Distinct Real Roots Inversion

2. Harmonic Oscillator with Complex Roots

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