What If We Started With the Inverse? — Kyungu's Approach

Author: Pathy Kyungu

Contact: leprofesseurkyungu@gmail.com

1. Introduction

In traditional mathematics teaching, students first learn derivatives before integrals, direct transforms before inverse transforms, and abstract theory before applications. But is this sequence the most natural? What if starting with inverses could lead to deeper understanding?

This article introduces a pedagogical framework based on the Kyungu Formula, suggesting that we should teach the inverse Laplace transform first, before introducing the direct transform as a consequence.

2. Kyungu Formula for Inverse Laplace Transform

The Kyungu Formula expresses the inverse Laplace transform of a function \( F(p) \) as a series, without requiring contour integrals:

\[ \mathcal{L}^{-1}[F](t) = \varphi(0)\cdot \delta(t) + \sum_{n=1}^\infty \frac{\varphi^{(n)}(0)}{(n!)^2} \cdot n \cdot t^{n-1} \]

with the auxiliary function:

\[ \varphi(x) = F\left( \frac{1}{x} \right) \]

\( \delta(t) \) denotes the Dirac delta distribution (impulse at \( t=0 \)).
\( \varphi^{(n)}(0) \) is the \(n\)-th derivative of the auxiliary function evaluated at zero.

3. Extension to the Inverse Mellin Transform

Using the logarithmic substitution \( t = -\ln(x) \), the formula extends naturally to the inverse Mellin transform:

\[ \mathcal{M}^{-1}[F](x) = \varphi(0)\cdot \delta(x-1) + \sum_{n=1}^\infty \frac{\varphi^{(n)}(0)}{(n!)^2} \cdot n \cdot \left( \ln\frac{1}{x} \right)^{n-1} \]

This expression is known as the Kyungu–Mellin–Laplace (KML) Formula.

4. Pedagogical Interest

Introduce the inverse Laplace transform first.
Present the direct Laplace transform as a derived concept.
Focus learning on reconstructing signals \( f(t) \) from known transforms.
Use derivatives and series as central analytical tools.

5. Philosophical Reflection

"We used to teach the Laplace transform from left to right. But that was before we discovered how powerful the inverse could be."

6. Conclusion

The Kyungu Formula is not only a mathematical tool but also a new educational paradigm, emphasizing inversion, signal recovery, and analytical thinking through simple series expansions.