Laplace Transform Visualizer

This tool makes the Laplace transform visible as a signed area integral.

The three plots

• Left - \(f(t)\): the original function with the decaying weight \(e^{-st}\) overlaid (dashed).
• Centre - integrand \(f(t)\cdot e^{-st}\): green areas add to \(F(s)\); red areas subtract.
• Right - \(F(s)\): the transform as a function of \(s\). The blue dot marks the current \(s\).

Convolution pairs

When a convolution pair \(f * g\) is selected, the visualized function is \(h(t)=(f*g)(t)\) and the transform shown is \(H(s)=F(s)\cdot G(s)\), illustrating the Convolution Theorem.

Numerical method - Gauss–Laguerre quadrature

The transform value \(F(s)\) is computed directly on the half-line with Gauss–Laguerre quadrature, which builds the \(e^{-st}\) weight into the rule. The substitution \(t=u/s\) places the nodes correctly for every \(s\):

There is no upper limit to choose: the infinite tail is handled analytically by the weight, so no truncation is involved. The nodes \(x_k/s\) move with \(s\), which keeps the sampling aligned with where the integrand actually lives.

Gauss–Laguerre assumes the smooth factor \(f(u/s)\) is analytic; a jump or kink — a Heaviside step \(u_a(t)\), or the corner of \(u_a(t)*1\) — destroys that and drops the accuracy to first order. So when \(f\) is non-smooth at a point \(a\), we split the integral there rather than let a node straddle it: the tail is shifted with \(t=a+v/s\) so the break lands at an endpoint, \(\int_a^\infty\! e^{-st}f(t)\,dt=\dfrac{e^{-as}}{s}\sum_k w_k\,f(a+x_k/s)\), with Gauss–Legendre on any finite panels before it. That restores full accuracy on the step examples.

A rapidly oscillating factor, \(\sin(\omega t)\) or \(\cos(\omega t)\), breaks the analytic assumption in a different way: once \(\omega\gtrsim s\) there are many wiggles between Laguerre nodes and the accuracy collapses (at \(s=0.5,\ \omega=4\) the plain rule is off by more than \(100\%\)). For these we switch to the Ooura–Mori double-exponential formula for Fourier integrals: writing \(e^{-st}f(t)\) as a sum of \(q(t)e^{-\lambda t}\{\sin,\cos\}(\omega t)\) pieces, each oscillatory piece \(\int_0^\infty g(x)\sin(\omega x)\,dx\) is evaluated with the substitution \(\omega x = M\varphi(t)\), whose nodes march toward the zeros \(k\pi/\omega\) so the transformed integrand decays double-exponentially. Accuracy then holds up as \(\omega\) grows — back to machine precision on the oscillatory examples.

Custom \(f(t)\)

Pick Custom from the function menu to type your own \(f(t)\) (in \(t\), e.g. 1/(1+t^2), exp(-t)sin(3t), sqrt(t)). Since an arbitrary \(f\) has no closed-form transform, \(F(s)\) is computed purely by the Gauss–Laguerre quadrature above and the \(F(s)\) plot traces that numerical value — the exact-value and relative-error cells are hidden, because there is nothing to compare against. Convergence needs \(f(t)e^{-st}\to0\); if a value looks unstable, raise \(s\).

Shifting theorems

• First (\(s\)-shift): \(\mathcal{L}\{e^{at}f(t)\} = F(s-a)\).
• Second (\(t\)-shift): \(\mathcal{L}\{u_c(t)f(t-c)\} = e^{-cs}F(s)\).

Convolution theorem

• \(\mathcal{L}\{(f*g)(t)\} = F(s)\cdot G(s)\) where \((f*g)(t)=\int_0^t f(\tau)g(t-\tau)\,d\tau\).