Hypernumbers and Extrafunctions: Extending the Classical Calculus

“Hypernumbers and Extrafunctions” presents a rigorous mathematical approach to operate with infinite values. First, concepts of real and complex numbers are expanded to include a new universe of numbers called hypernumbers which includes infinite quantities. This brief extends classical calculus based on real functions by introducing extrafunctions, which generalize not only the concept of a conventional function but also the concept of a distribution. Extrafucntions have been also efficiently used for a rigorous mathematical definition of the Feynman path integral, as well as for solving some problems in probability theory, which is also important for contemporary physics.

This book introduces a new theory that includes the theory of distributions as a subtheory, providing more powerful tools for mathematics and its applications. Specifically, it makes it possible to solve PDE for which it is proved that they do not have solutions  in distributions. Also illustrated in this text is how this new theory allows the differentiation and integration of any real function. This text can be used for enhancing traditional courses of calculus for undergraduates, as well as for teaching a separate course for graduate students.

Encountering problems that they were not able to solve, mathematicians, as a rule, introduced new structures, extending the existing ones and making possible to solve previously unsolvable problems. For instance, irrational, real and complex numbers were introduced to make possible solving all algebraic equations. The stimulus to develop the theory of hypernumbers and extrafunctions comes from physics. Exploring mysteries of the microworld, physicists often encounter situations when their formulas acquire infinite values due the divergence of the used series and integrals. At the same time, all measured values are naturally finite. A popular way to eliminate this discrepancy between theory and experiment was an artificial manipulation with formulas, which often allowed to get rid of infinite values. Another natural way to deal with such situation is to learn how to rigorously work with infinities and come to finite values given by measurements. Mathematicians suggested several approaches to a rigorous operation with infinite values by introducing infinite numbers. The most popular of them are: transfinite numbers of Cantor, nonstandard analysis of Robinson, and surreal numbers. However, all these constructions, which contributed a lot to the development of mathematics, especially, Cantor 's set theory, brought very little to the realm of physics. However, there is a mathematical theory called the theory of distributions, which allowed physicists to rigorously operate with functions that take infinite values. An example of such a function is the Heaviside-Dirac 's delta-function. Although at first physicists were skeptical about this theory, now it has become one of the most efficient tools of theoretical physics. In this book, we represent another rigorous mathematical approach to operation with infinite values. At first, the concepts of real and complex numbers are extended in such a way that the new universe of numbers called hypernumbers includes infinite quantities. It is necessary to remark that in contrast to nonstandard analysis, there are no infinitely small hypernumbers. This is more relevant to the situation in physics, where infinitely big values emerge from theoretical structures but physicists have never encountered infinitely small values. The next step of extending the classical calculus based on real and complex functions is introduction of extrafunctions, which generalize not only the concept of a conventional function but also the concept of a distribution. This made possible to solve previously unsolvable problems. For instance, there are linear partial differential equations for which it is proved that they do not have solutions not only in conventional functions but even in distributions. At the same, it is proved that all these and many other equations have solutions in extrafunctions. Extrafucntions have been also efficiently used for a rigorous mathematical definition of the Feynman path integral, as well as for solving some problems in probability theory, which is also important for contemporary physics.