A transition integral transform obtained from generalization of the Fourier transform

Abstract

We introduce a generalized integral transform (GIT) whose integration path lies on the complex plane. The GIT has both bilateral and unilateral versions, and generalizes a set of known integral transforms, e.g. Fourier, Laplace, allowing to solve integro-differential equations with initial conditions in a more efficient way. We discuss the inversion formula of the new transform and give some particular examples of its application for solving differential and integral equations. The basic properties of the new integral transform are discussed in detail. Keywords: Integral transforms, Fourier transform, Differential equations, Fractional calculus