Integral Transformation and Different Forms

Abstract

The main problem under study is the construction of the complete convergent series development of integral transforms. The proposed paper is a study on integral transformation and its different forms. It also studies the relation between integral transformations and different forms. Integral transform, mathematical operator that produces a new function f(y) by integrating the product of an existing function F(x) and a so-called kernel function K(x, y) between suitable limits. The process, which is called transformation, is symbolized by the equation f(y) = ∫K(x, y)F(x)dx. Several transforms are commonly named for the mathematicians who introduced them: in the Laplace transform, the kernel is e−xy and the limits of integration are zero and plus infinity; in the Fourier transform, the kernel is (2π)−1/2e−ixy and the limits are minus and plus infinity.