Laplace Transform

The Laplace transform converts a time-domain function f(t) into a complex-frequency-domain function F(s):

$F(s) = \mathcal{L}\{f(t)\} = \int_0^{\infty} f(t)\, e^{-st}\, dt$

It is one of the most powerful tools in mathematical physics, turning differential equations into algebraic ones.

Common Transform Pairs

The Convolution Theorem

If $F(s) = \mathcal{L}\{f\}$ and $G(s) = \mathcal{L}\{g\}$, then:

$\mathcal{L}\{f * g\} = F(s)\cdot G(s)$

where the convolution is $(f*g)(t) = \int_0^t f(\tau) g(t-\tau)\, d\tau$

This makes solving linear ODEs with initial conditions straightforward: apply the transform, solve the algebraic equation, then invert.

Numerical Laplace Transform

For functions without a closed-form transform, we approximate:

$F(s) \approx \sum_{i=0}^{N-1} f(t_i)\, e^{-s\, t_i}\, \Delta t$

where $t_i = i \cdot \Delta t$ and $\Delta t = T_{\max}/N$. This rectangle-rule quadrature works well when f(t) decays fast enough that the tail beyond $T_{\max}$ is negligible.

Implementation

Implement the following functions:

• laplace_exp(a, s) — analytic: returns $1/(s-a)$
• laplace_sin(omega, s) — analytic: returns $\omega/(s^2+\omega^2)$
• laplace_cos(omega, s) — analytic: returns $s/(s^2+\omega^2)$
• laplace_tn(n, s) — analytic: returns $n!/s^{n+1}$
• laplace_numerical(f_func, s, T_max=20, N=10000) — numerical rectangle-rule approximation

import math

def laplace_exp(a, s):
    return 1.0 / (s - a)

def laplace_sin(omega, s):
    return omega / (s**2 + omega**2)