# 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. > Canonical page: https://elysiatools.com/en/tools/inverse-laplace-calculator - **Category:** Math & Numbers - **Keywords:** inverse laplace transform, partial fractions, rational function, laplace pairs, durand-kerner, complex roots, repeated roots, ode solving, transfer function, calculus ## Overview The Inverse Laplace Transform Calculator converts proper rational functions F(s) = N(s)/D(s) back into time-domain expressions f(t) using partial fraction decomposition. It automatically identifies real, repeated, and complex conjugate denominator roots, solves the linear coefficient system, and applies standard inverse Laplace pairs term by term. ## Inputs - **Numerator Coefficients N(s) (constant first)** (text): Comma-separated numerator coefficients, constant term first, e.g. 5,3 → 3s + 5. - **Denominator Coefficients D(s) (constant first)** (text): Comma-separated denominator coefficients, constant term first, degree ≤ 6, e.g. 2,3,1 → s² + 3s + 2. - **Decimal Places** (number) ## When to use - Solving linear ordinary differential equations (ODEs) converted into algebraic s-domain expressions. - Evaluating the time-domain impulse or step response of transfer functions in control systems and signal processing. - Decomposing proper rational functions with real, repeated, or complex roots into standard exponential and trigonometric components. ## How it works - Enter the numerator N(s) and denominator D(s) polynomial coefficients as comma-separated values, starting with the constant term (degree 0). - The solver computes the roots of the denominator polynomial (up to degree 6) and categorizes them into real roots, repeated multiplicities, and complex conjugate pairs. - A system of linear equations determines the partial fraction coefficients, and the calculator applies term-by-term inversion to generate the time-domain function f(t). ## Use cases - Control engineering: Deriving analytical time responses from closed-loop s-domain transfer functions. - Circuit theory: Calculating transient voltages and currents in RLC network differential models. - Engineering coursework: Verifying intermediate partial fraction steps and final inverse Laplace solutions. ## Frequently asked questions ### 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? 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