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Lab 02

Transform Lab

How do a kernel and a region of convergence turn a function into its transform?

Laplace, Fourier, and Mellin modes for a small table of classical pairs. Each mode plots the input, the kernel, and their product, shows the transform with its region or strip of convergence, and compares the closed form with numerical integration.

  • Computational Demonstration
  • Classical Foundation
∫ f(t) e⁻ˢᵗ dt

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Scope of this instrument

Supported

  • Laplace, F(s)=∫0∞f(t)e−st dtF(s)=\int_0^\infty f(t)e^{-st}\,dt for real s>−αs>-\alpha: e−αte^{-\alpha t}, e−αtcos⁡ωte^{-\alpha t}\cos\omega t, e−αtsin⁡ωte^{-\alpha t}\sin\omega t, and tne−αtt^n e^{-\alpha t} with 0≤n≤60 \le n \le 6.
  • Fourier, with the convention F(ω)=∫−∞∞f(t)e−iωt dtF(\omega)=\int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt: e−a∣t∣↔2a/(a2+ω2)e^{-a|t|} \leftrightarrow 2a/(a^2+\omega^2) and e−t2/(2σ2)↔σ2π e−σ2ω2/2e^{-t^2/(2\sigma^2)} \leftrightarrow \sigma\sqrt{2\pi}\,e^{-\sigma^2\omega^2/2}.
  • Mellin, F(s)=∫0∞xs−1f(x) dxF(s)=\int_0^\infty x^{s-1}f(x)\,dx on real ss: e−x↔Γ(s)e^{-x} \leftrightarrow \Gamma(s) for s>0s>0, and 1/(1+x)↔π/sin⁡(πs)1/(1+x) \leftrightarrow \pi/\sin(\pi s) for 0<s<10<s<1.
  • Numerical checks by adaptive Gauss–Kronrod quadrature, with the truncation point reported.

Not supported

  • There is no general symbolic solver: only the listed pairs are available.
  • Complex ss or ω\omega, inverse transforms, and distributional transforms.
  • Points within 0.050.05 of the abscissa or strip edge, where the integral converges too slowly for a reliable check.
  • The book's operator-based derivations (Ch. 2) are described, not executed.

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