The Natural Logarithmic Transformation and its Applications

In this paper, a new integral transformation is proposed, where the transformation kernel is the natural logarithmic function l(n)(proportional to x), proportional to > 0 x > 0, the transformation interval is the closed interval [1/proportional to, 1], and the range of its kernel l(n)(proportional to x) is the entire set of the real numbers (-infinity < ln (proportional to x)< infinity). The wide range of the kernel for the proposed transformation giving it a wider usage from the other transformations such as Laplace transformation which its kernel is (e(-ax))and its range includes the natural numbers only ((e(-ax)) > 0). The proposed integral transformation is called "the logarithmic integral transformation" based on the kernel of the transformation. Some properties and theorems are presented for this new transformation.