Laplace Transform Calculator

This Laplace transform calculator works in both directions. In forward mode, type a function of t such as t^2 + 3sin(2t), e^(−2t)cos(3t) or t·sin(t) and it splits it into standard pieces — powers of t, exponentials, sines, cosines, sinh and cosh, and their products — and applies the table of transforms and the shift and multiplication-by-tⁿ theorems to return F(s), the region of convergence and a numerical check of the integral.

In inverse mode, type a rational function of s such as (s + 3)/(s^2 + 4s + 13) or 1/(s(s + 1)^2). The calculator finds the poles of the denominator, computes the partial-fraction (residue) coefficients — including repeated and complex poles — and writes f(t) as a combination of exponentials, powers of t, sines and cosines.

It is built for differential equations, signals and control-systems students who need to check a transform from a table, and for engineers solving circuit or vibration problems. Inverse transforms are computed numerically from the poles, so coefficients are shown as fractions when they are simple and as decimals otherwise. Use t as the variable in forward mode and s in inverse mode.

The Laplace transform turns a function of time into a function of the complex frequency s. Because it is linear, a sum is transformed term by term. The core table entries are L{1} = 1/s, L{tⁿ} = n!/sⁿ⁺¹, L{sin bt} = b/(s² + b²), L{cos bt} = s/(s² + b²), L{sinh bt} = b/(s² − b²) and L{cosh bt} = s/(s² − b²). Multiplying by e^(at) shifts s to s − a (first shifting theorem), and multiplying by tⁿ gives (−1)ⁿ times the n-th derivative of F(s). Products of sines and cosines are first rewritten with product-to-sum identities.

For the inverse transform, a proper rational function N(s)/D(s) is expanded in partial fractions. Each pole p of multiplicity m contributes Σ Aⱼ tʲ⁻¹ e^(pt)/(j − 1)!, where the coefficients come from the Taylor expansion of (s − p)ᵐ F(s) at p (the residue method). A pair of complex poles α ± βi combines into e^(αt)(C cos βt + D sin βt).