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In the present article the author extends the Fourier transform to a moregeneral class of functions; First to power-law functions with integer andhalf-integer exponents then to the widely used quantum statistics function(Fermi-Dirac and Bose-Einstein-distributions). The results are used to deriverather straightforwardly an integral formula. Moreover, generalized Fouriertransforms are used to define fractional derivatives of distributions(generalized functions) and Fourier series (periodic functions) as well.

Generalized Fourier transforms applied to fractional derivatives and Quantum Statistics Distributions