Laplace Transform Visualizer

See \(\mathcal{L}\{f(t)\}(s)\) as a signed area — watch the integrand accumulate, track convergence, and read the exact formula.

\[\mathcal{L}\{f(t)\}(s) \;=\; F(s) \;=\; \int_0^{\infty} f(t)\,e^{-st}\,dt\]
Function \(f(t)\)
\(s = 1.00\)
T = 8
\(\displaystyle\int_0^T\!\max(g,0)\,dt\)
—
positive contribution
\(-\)
\(\displaystyle\int_0^T\!|\min(g,0)|\,dt\)
—
negative contribution
\(=\)
\(\displaystyle\int_0^T\!f(t)e^{-st}\,dt\)
—
numerical net area
\(\approx\)
Exact \(F(s)\)
—
closed-form value
\(|\)
Relative error
—
\(\bigl|\tfrac{\mathrm{num}-F(s)}{F(s)}\bigr|\)
\(f(t)\) — original function
\(f(t)\) \(e^{-st}\) weight
\(f(t)\cdot e^{-st}\) — integrand  (net signed area \(=F(s)\))
Adding to \(F(s)\) Subtracting from \(F(s)\) \(f(t)\cdot e^{-st}\)
\(F(s)\) — Laplace transform
\(F(s)\) current \(s\)
Select a function above to see its Laplace transform.
Convergence: \(\displaystyle A(T)=\int_0^{T}f(t)\,e^{-st}\,dt\;\longrightarrow\;F(s)\) as \(T\to\infty\)
Convolution Theorem

Convolution Theorem

\[\mathcal{L}\!\left\{\int_0^t f(\tau)\,g(t-\tau)\,d\tau\right\} \;=\; \mathcal{L}\{f*g\} \;=\; F(s)\cdot G(s)\]

If \(\mathcal{L}\{f\}=F(s)\) and \(\mathcal{L}\{g\}=G(s)\), then the transform of their convolution is the product of their individual transforms. Selecting a convolution pair below visualizes \(h(t)=(f*g)(t)\) and plots \(H(s)=F(s)\cdot G(s)\).

  • \(\mathcal{L}\{u_a(t)*1\} = \dfrac{e^{-as}}{s}\cdot\dfrac{1}{s} = \dfrac{e^{-as}}{s^2}\)
  • \(\mathcal{L}\{\sin(\omega t)*1\} = \dfrac{\omega}{s^2+\omega^2}\cdot\dfrac{1}{s} = \dfrac{\omega}{s(s^2+\omega^2)}\)
  • \(\mathcal{L}\{e^{at}*e^{bt}\} = \dfrac{1}{s-a}\cdot\dfrac{1}{s-b} = \dfrac{1}{(s-a)(s-b)}\quad(a\ne b)\)
  • \(\mathcal{L}\{e^{at}*\sin(\omega t)\} = \dfrac{1}{s-a}\cdot\dfrac{\omega}{s^2+\omega^2} = \dfrac{\omega}{(s-a)(s^2+\omega^2)}\)
Shifting Theorems

First Shifting Theorem (\(s\)-shift)

\[\mathcal{L}\{e^{at}f(t)\} = F(s-a)\]

Multiplying \(f\) by \(e^{at}\) shifts \(F(s)\) right by \(a\); the ROC becomes \(s>a\).

  • \(\mathcal{L}\{e^{at}\sin\omega t\} = \dfrac{\omega}{(s-a)^2+\omega^2}\)
  • \(\mathcal{L}\{e^{at}\cos\omega t\} = \dfrac{s-a}{(s-a)^2+\omega^2}\)
  • \(\mathcal{L}\{t^n e^{at}\} = \dfrac{n!}{(s-a)^{n+1}}\)

Second Shifting Theorem (\(t\)-shift)

\[\mathcal{L}\{u_c(t)f(t-c)\} = e^{-cs}F(s)\]

Delaying \(f\) by \(c\) (with step \(u_c\)) multiplies \(F(s)\) by \(e^{-cs}\).

  • \(\mathcal{L}\{u_c(t)\} = \dfrac{e^{-cs}}{s}\)
  • \(\mathcal{L}\{u_c(t)(t-c)\} = \dfrac{e^{-cs}}{s^2}\)
  • \(\mathcal{L}\{u_c(t)\sin\omega(t-c)\} = \dfrac{\omega e^{-cs}}{s^2+\omega^2}\)
\(f(t)\) \(\mathcal{L}\{f(t)\} = F(s)\) ROC Visualize