Laplace Transform Table Calculator: Transforms and Inverse Transforms of Standard Functions

The standard Laplace pairs, evaluated at a point.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose the transform pair and whether you want the transform or the inverse.

Enter the constants K, a, w and the power n that the pair uses.

Enter the value of s or t to evaluate at.

Formulas used

  • F(s) = \int_0^{\infty} f(t)e^{-st}\,dt
  • \mathcal{L}\{e^{-at}\} = \frac{1}{s + a}
  • \mathcal{L}\{\sin\omega t\} = \frac{\omega}{s^2 + \omega^2}

Worked example

With the default numbers in the calculator above:

  1. The function of time: Decaying exponential: f(t) = K e^(-a t), F(s) = K / (s + a)
  2. Its Laplace transform: See the formula (L{f(t)} = F(s))
  3. F(s) at the value of s you entered: 0.166667

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Transform pair Sine: f(t) = K sin(w t), F(s) = K w / (s^2 + w^2); Angular frequency w 2 rad/s; Value of s (transform) or of t (inverse) 3.

  1. The function of time: Sine: f(t) = K sin(w t), F(s) = K w / (s^2 + w^2)
  2. Its Laplace transform: See the formula (L{f(t)} = F(s))
  3. F(s) at the value of s you entered: 0.153846

Common questions

What is the Laplace transform used for?

It turns a differential equation in time into an algebra problem in s. It is used for circuits, control systems and vibrations.

How do I invert a complicated F(s)?

Split it into partial fractions so each term matches a standard pair, then invert each term.

What is a pole?

A value of s that makes the bottom of F(s) zero. The poles decide how the time response behaves.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any engineering mathematics text (the Laplace transform table).

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