# The generalized integral Fourier transform

 1Virchenko, NO2Chetvertak, MO1NTU of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"2National Technical University of Ukraine "Kyiv Polytechnic Institute" Dopov. Nac. akad. nauk Ukr. 2015, 8:7-12 https://doi.org/10.15407/dopovidi2015.08.007 Section: Mathematics Language: Ukrainian Abstract:  The generalized integral Fourier transform \begin{gather*} \widetilde{f}(\alpha)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^{\infty} f(x)e^{i{\alpha}x}{_{1}\Phi^{\tau,\beta}_{1}}(a;c;-r({\alpha}x))\,dx, \end{gather*} where $\rm{Re}\,\it{c}> \rm{Re}\,\it{a}>0$, $({\tau,\beta})\subset R$, $\tau-\beta<1$, $r>0$, and  ${{_{1}\Phi}^{\tau,\beta}_{1}}(\ldots )$ is the $(\tau,\beta)$-confluent hypergeometric function, is introduced. The inversion formula of this integral transform is proved. The basic properties of a new integral Fourier transform and some examples are given. Keywords: confluent hypergeometric function, Fourier’ integral transform, generalized integral transform
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