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In mathematics, the '''N-transform''' is an [[integral transform]] similar to the [[Laplace transform]] and [[Sumudu transform]], introduced by Zafar Hayat Khan<ref name="Khan"/> 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<ref name="Belgacem3"/> has renamed it the '''natural transform''' and has proposed a detail theory and applications.<ref name="Belgacem"/><ref name="Belgacem2"/>

==Formal definition==
The natural transform of a [[function (mathematics)|function]] ''f''(''t''), defined for all real numbers ''t''&nbsp;≥&nbsp;''0'', is the function ''R''(''u'',&nbsp;''s''), defined by:

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

Khan<ref name="Khan"/> showed that the above integral converges to [[Laplace transform]] when ''u''&nbsp;=&nbsp;1, and into [[Sumudu transform]] for ''s''&nbsp;=&nbsp;1.

==See also==
* [[Laplace transform]]
* [[Sumudu transform]]
* [[List of transforms]]

==References==

{{reflist|refs=
<ref name="Khan">Khan, Z.H., Khan, W.A., "N-Transform-Properties and Applications" NUST Journal of Engineering Sciences 1(2008), 127–133.</ref>
<ref name="Belgacem3">Belgacem, F. B. M and Silambarasan, R. Theory of the Natural transform. Mathematics in Engg Sci and Aerospace (MESA) journal. Vol. 3. No. 1. pp 99–124. 2012.</ref>
<ref name="Belgacem">Belgacem, F. B. M., and R. Silambarasan. "Advances in the Natural transform." AIP Conference Proceedings. Vol. 1493. 2012.</ref> http://en.wikipedia.org/w/index.php?title=N_Transform&action=submit#editform
<ref name="Belgacem2">Silambarasan, R., and F. B. M. Belgacem. "Applications of the Natural transform to Maxwell’s Equations." 12–16.</ref>
}}

[[Category:Integral transforms]]


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