The Laplace Transformation of Distributions

The physicist needs to introduce certain concepts which enable him to mathematically describe physical phenomena. One of these concepts is the “impulse” δ which is supposed to mathematically represent a shock like, for instance, the impact of a hammer in mechanics, or a large voltage increase of exceedingly short duration in electrical engineering. Concepts of this nature cannot be comprehended within the frame of the classical theory of functions. However the modem theory of distributions embraces these concepts in a consistent manner. Moreover, this new theory avoids many of the difficulties of classical analysis. Thus, we must extend the theory of the Laplace-transformation, which in the previous Chapters has been developed for functions only, to distributions. This necessitates a knowledge of the foundations of the theory of distributions which is understood here in the sense of L. Schwartz. The termini and theorems of the theory of distributions employed here are compiled and presented in an Appendix, organized in 22 statements; in the text we shall refer to this Appendix whenever desirable, citing App. and the No. of the statement of specific interest.