# N-transform

In mathematics, the Natural transform is an integral transform similar to the Laplace transform and Sumudu transform, introduced by Zafar Hayat Khan[1] in 2008. It converges to both Laplace and Sumudu transform just by changing variables. Given the convergence to the Laplace and Sumudu transforms, the N-transform inherits all the applied aspects of the both transforms. Most recently, F. B. M. Belgacem[2] has renamed it the natural transform and has proposed a detail theory and applications.[3][4]

## Formal definition

The natural transform of a function f(t), defined for all real numbers t ≥ 0, is the function R(us), defined by:

$R(u, s) = \mathcal{N}\{f(t)\} = \int_0^\infty f(ut)e^{-st}\,dt.\qquad(1)$

Khan[1] showed that the above integral converges to Laplace transform when u = 1, and into Sumudu transform for s = 1.