Inverse Laplace Transform Calculator

This tool inverts a Laplace transform in the two cases it can do honestly: a match against a fixed table of eleven standard pairs, and a proper fraction whose denominator is already written as two or three different real linear factors. Pick the shape, type the constants, and it writes F(s) back with your numbers in it and returns f(t), the partial fraction residues where they apply, and the value of f at a t you choose. The residues come from Heaviside cover-up, the same method behind the partial fraction decomposition tool on this site, and each residue over one factor inverts to that residue times an exponential. There is no computer algebra here, so the page will not factor a polynomial for you, and it refuses repeated roots outside the 1/(s - a)^n entry, irreducible quadratic factors mixed with linear ones, complex roots and general time delays beyond the unit step.

Calculator Type

Pick the shape of F(s) from the table, type the constants in it, and read off f(t).

The whole of F(s) is multiplied by this, and so is f(t). Leave it at 1 for the bare table entry.
Used by the 1/(s - a), 1/(s - a)^n, e^(at) sin, e^(at) cos, sinh and cosh forms.
Used by the four sine and cosine forms. It must not be zero there.
Used by the 1/s^n and 1/(s - a)^n forms only.
Used by the delayed unit step only.
Optional. The symbolic answer does not depend on it; this only puts a number to it.
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For F(s) = (a s + b) / ((s - r1)(s - r2)) with r1 and r2 different real numbers. Cover-up gives the residues and each term inverts to an exponential.

Set this to zero for a constant numerator.
The numerator is a s + b.
The value of r in the factor (s - r): enter 2 for (s - 2) and -3 for (s + 3).
The value of r in the factor (s - r): enter 2 for (s - 2) and -3 for (s + 3).
Optional. The symbolic answer does not depend on it; this only puts a number to it.
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The same method one factor further: F(s) = (a s^2 + b s + c) / ((s - r1)(s - r2)(s - r3)), all three roots different and real.

Set this to zero for a numerator of lower degree.
The numerator is a s^2 + b s + c.
The numerator is a s^2 + b s + c.
The value of r in the factor (s - r): enter 2 for (s - 2) and -3 for (s + 3).
The value of r in the factor (s - r): enter 2 for (s - 2) and -3 for (s + 3).
The value of r in the factor (s - r): enter 2 for (s - 2) and -3 for (s + 3).
Optional. The symbolic answer does not depend on it; this only puts a number to it.
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