An integral analogue of Taylor’s series and its use in computing Fourier transforms

Abstract:

In this paper, an integral analogue of Taylor’s series \[ f(z) = \int _{ - \infty }^\infty {{D^\omega }f({z_0})} {(z - {z_0})^\omega }/\Gamma (\omega + 1)d\omega \] is discussed. ${D^\omega }f(z)$ is a fractional derivative of order $\omega$. Extensions of this integral are also given, one of which is an integral analogue of Lagrange’s expansion. These integrals are shown to be generalizations of the Fourier integral theorem. Several special cases of these integrals are computed, and a table of Fourier transforms emerges.