Author: Pathy Kyungu
Publication Date: July 20, 2025
Version: 1.0 – Official Published Version
The Kyungu Formula is applied here to evaluate the inverse Laplace transform of the function:
\[ F(p) = \arctan\left(\frac{1}{p^2}\right) \]
The method leads to both a series representation and a closed-form expression involving hyperbolic and trigonometric functions.
The formula expresses the inverse Laplace transform of a function \( F(p) \) as follows:
\[ \mathcal{L}^{-1}[F](t) = \varphi(0) \cdot \delta(t) + \sum_{n=1}^{\infty} \frac{\varphi^{(n)}(0)}{(n!)^2} \cdot n \cdot t^{n-1} \]
with the auxiliary function:
\[ \varphi(p) = F\left(\frac{1}{p}\right) \]
Starting from:
\[ F(p) = \arctan\left( \frac{1}{p^2} \right) \]
we get:
\[ \varphi(p) = \arctan(p^2) \]
The series expansion of \(\arctan(x)\) is:
\[ \arctan(x) = \sum_{k=0}^\infty (-1)^k \cdot \frac{x^{2k+1}}{2k+1} \]
Therefore:
\[ \varphi(p) = \sum_{k=0}^\infty (-1)^k \cdot \frac{p^{4k+2}}{2k+1} \]
The derivatives at zero are given by:
\[ \varphi^{(4k+2)}(0) = (-1)^k \cdot \frac{(4k+2)!}{2k+1} \]
Thus, the series expression becomes:
\[ \mathcal{L}^{-1}[F](t) = 2 \sum_{k=0}^\infty (-1)^k \cdot \frac{t^{4k+1}}{(4k+2)!} \]
This infinite sum simplifies into the following exact closed-form expression:
\[ \frac{2}{t} \cdot \sinh\left( \frac{t\sqrt{2}}{2} \right) \cdot \sin\left( \frac{t\sqrt{2}}{2} \right) \]
Therefore:
\[ \mathcal{L}^{-1}\left[ \arctan\left( \frac{1}{p^2} \right) \right](t) = \frac{2}{t} \cdot \sinh\left( \frac{t\sqrt{2}}{2} \right) \cdot \sin\left( \frac{t\sqrt{2}}{2} \right) \]
This result does not appear in classical Laplace transform tables. It demonstrates the analytical power of the Kyungu Formula.
Official Website: pathykyungu.github.io
This draft is intended to inspire the writing of a future Wikipedia article by independent contributors.