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Theory and Modern Applications

Table 1 Integral transform of some basic functions

From: New general integral transform via Atangana–Baleanu derivatives

f(t)

\(T \{ f(t),s \} =\mathcal{F}_{\mathrm{J}}(s)\)

c

\(\frac{\varphi (s)}{\psi (s)}c\), \(c\in \mathbb{R} \)

t

\(\frac{\varphi (s)}{\psi (s)^{2}}\)

\(t^{\alpha }\)

\(\frac{\Gamma (\alpha +1)\varphi (s)}{\psi (s)^{\alpha +1}}\), α>0

sint

\(\frac{\varphi (s)}{\psi (s)^{2}+1}\)

cost

\(\frac{\varphi (s)\psi (s)}{\psi (s)^{2}+1}\)