What is the Laplace transform?

The (one-sided) Laplace transform converts a time-domain function f(t) (for t ≥ 0) into an s-domain function F(s) using an exponentially decaying “weight”:

You can think of it as a fancy way of summarizing how f(t) behaves over time. The term e^(−st) forces the integral to converge when s is large enough. That “large enough” requirement shows up as the region of convergence (ROC), often written as Re(s) > a for some constant a.

• Derivatives become algebra: Laplace turns differential equations into simpler algebraic equations.
• Initial conditions fit naturally: The one-sided Laplace transform bakes in f(0), f′(0), etc.
• Time shifts are easy: Multiplying by e^(−cs) represents delayed signals.
• Great for LTI systems: Poles/zeros and transfer functions live in the s-domain.

• 𝓛{1} = 1/s
• 𝓛{t^n} = n! / s^{n+1} (for integer n ≥ 0, ROC Re(s) > 0)
• 𝓛{e^{at}} = 1/(s − a) (ROC Re(s) > a)
• 𝓛{sin(ωt)} = ω/(s² + ω²) (ROC Re(s) > 0)
• 𝓛{cos(ωt)} = s/(s² + ω²) (ROC Re(s) > 0)
• 𝓛{f(t−c)u(t−c)} = e^{−cs}F(s) (time-shift theorem)

Examples you can copy

These are the exact patterns the calculator uses (so your results match).

Why the e^{6}? Because for t ≥ 3, e^{2t} = e^{2(t−3)}·e^{6}. This calculator shows that factor explicitly so you see where constants come from.

What this calculator is doing (behind the scenes)

A truly “general” Laplace transform engine would need symbolic integration, simplification, and branch handling. That’s heavy. For virality and usefulness, this calculator focuses on the most common table forms students actually use in class. It follows a simple flow:

You select a function type like t^n or sin(ωt). Each type corresponds to a known transform identity.

Your n, a, ω, and shift c get substituted into the identity.

Every identity comes with a simple condition like Re(s) > a or Re(s) > 0. Many homework mistakes happen by ignoring this, so we display it every time.

If you enter a value for s, we plug it into the formula and show a numeric value. This is useful for quick sanity checks, plotting, or verifying a transfer function at a particular frequency.

• We use s (standard in control/systems).
• Re(s) is the real part of s. In many intro classes, you can read it as “s must be bigger than …”.
• u(t−c) is the Heaviside unit step (0 before c, 1 after).

Frequently Asked Questions

• Is this the one-sided or two-sided Laplace transform?

  This calculator uses the one-sided (unilateral) Laplace transform: ∫₀^∞ e^{−st} f(t) dt. That’s the version used most often in differential equations and control systems with initial conditions at t = 0.

• Why does the answer include “Re(s) > …”?

  The integral converges only when the exponential weight e^{−st} decays fast enough. That’s the region of convergence (ROC). Many tables assume it silently — we show it so you don’t forget.

• What if my function is a sum, like 3e^{2t} + 4sin(5t)?

  Use linearity: 𝓛{af(t) + bg(t)} = aF(s) + bG(s). Compute each part using this calculator (or a table), then add the results.

• What about derivatives and differential equations?

  A classic identity is 𝓛{f′(t)} = sF(s) − f(0) (and higher derivatives follow a pattern). This page focuses on base functions because that’s where most students get stuck. Once you have F(s), you can apply derivative rules to solve ODEs.

• Does this support piecewise functions?

  Partially. The shift options with u(t−c) cover the most common piecewise form: “turning on” a function at time c. For more complex piecewise definitions, you typically rewrite them using step functions, then apply linearity and the time-shift theorem.

• Can ω be negative?

  Yes. For sin(ωt) and cos(ωt), negative ω changes sign for sine but not for cosine. This calculator preserves the sign you enter in the final expression. Many textbooks assume ω ≥ 0, so you may want to rewrite using absolute values if your course expects that.

• Why isn’t there a “type any expression” box?

  Doing that well requires a symbolic engine and careful branch handling. The goal here is speed and reliability for the transforms that show up constantly in engineering and math classes.