SIMPLIFYING AND EXTENDING A USEFUL CLASS OF SIGNALS AND IMPULSE RESPONSES

Mathematics is an essential tool for studying science and engineering, and calculus is one of the most important branches of mathematics for engineering. In this paper a new formula for evaluating integral x(n) e(ax) dx and a more generally applicable extension to polynomials are developed. This new approach illustrates the intimate relationship be tween differentiation and integration, and is simple enough for a fres hman taking the first course in calculus to derive it. Although a clos ed-form expression for this integral exists, it is cumbersome and rela tively more difficult to remember than the forms proposed in the paper . Also, the proposed formula readily generalizes to a larger class of polynomials, thus becoming much more useful. We show that these formul ae are particularly important for the analysis and use of a broad clas s of signals commonly encountered in the classroom and in practical si tuations. The proposed formulae are applied to Fourier Series, Fourier Transforms, LaPlace Transforms, and time domain convolution.