By Pathy Kyungu — Mathematician and Educator
The inverse Laplace transform is fundamental in the analysis of dynamic systems, differential equations, and signal processing.
Kyungu’s formula offers a new and elegant method to compute the inverse Laplace transform of \( F(p) \), avoiding complex integrals and classical tables.
Given a Laplace function \( F(p) \), define the auxiliary function:
\[ \varphi(x) = F\left(\frac{1}{x}\right) \]The inverse Laplace transform is then given by the series:
\[ f(t) = \varphi(0) \cdot \delta(t) + \sum_{n=1}^\infty \frac{\varphi^{(n)}(0)}{(n!)^2} \cdot n \cdot t^{n-1} \]Here:
\[ \varphi(x) = x^2 \]The only non-zero derivative is:
\[ \varphi''(0) = 2 \]The formula gives:
\[ f(t) = \frac{2}{(2!)^2} \cdot 2 \cdot t^1 = t \]Which corresponds to the known inverse Laplace transform of \( F(p) = \frac{1}{p^2} \).
Define:
\[ \varphi(x) = \frac{x^3}{x^3 + 1} \]Series expansion:
\[ \varphi(x) = \sum_{k=1}^\infty (-1)^{k+1} \cdot x^{3k} \]Derivatives at zero:
\[ \varphi^{(3k)}(0) = (-1)^{k+1} \cdot (3k)! \]Kyungu’s formula gives:
\[ f(t) = \sum_{k=1}^\infty \frac{(-1)^{k+1} \cdot 3k}{(3k)!} \cdot t^{3k - 1} \]Although \( F(p) = \frac{1}{1 + p^3} \) does not appear in classical tables, it is possible to recover a closed form using the cube roots of unity:
\[ 1 + p^3 = (p + 1)(p + \omega)(p + \omega^2), \quad \omega = e^{2i\pi/3} \]The inverse transform becomes:
\[ f(t) = \frac{1}{3}e^{-t} + \frac{1}{3\omega^2}e^{-\omega t} + \frac{1}{3\omega}e^{-\omega^2 t} \]Taking the real part:
\[ \boxed{ f(t) = \frac{1}{3} e^{-t} + \frac{2}{3} e^{t/2} \cdot \cos\left( \frac{\sqrt{3}}{2}t - \frac{\pi}{3} \right) } \]The series obtained via Kyungu’s formula converges towards this exact solution.
Kyungu’s formula provides an efficient and educational method to compute inverse Laplace transforms:
Last updated: July 2025 • Written by Pathy Kyungu