Laplace Transform Calculator

Standard Laplace transform table

The Laplace transform rewrites a time-domain function f(t) as a function of the complex frequency variable s:

F(s) = L{f(t)} = ∫[0,∞) e^(-st) f(t) dt

This calculator focuses on the table entries students and engineers use constantly:

The coefficient field multiplies the selected table entry. The optional offset b adds a constant term to the function, so it contributes b / s to the transform.

Worked example: transform t^2

For f(t) = t^2, choose the power form, set c = 1, and set n = 2.

The power rule is:

L{t^n} = n! / s^(n+1)

Substitute n = 2:

L{t^2} = 2! / s^3 = 2 / s^3

That is why the default example returns F(s) = 2 / s^3.

Common pitfalls

• Using the wrong parameter. In sin(at) and cos(at), a is the angular frequency inside the trig function, not an outside multiplier.
• Forgetting the coefficient. 5t^2 transforms to 10 / s^3, because the coefficient 5 multiplies 2! / s^3.
• Treating the tool as a full CAS. This calculator applies selected table rules. It does not simplify arbitrary expressions such as products, convolutions or piecewise functions.
• Ignoring convergence. Exponential transforms depend on the real part of s; for e^(at), the basic condition is s > a.