Hypernumbers and Extrafunctions

Extending the Classical Calculus

  • Mark Burgin

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-vii
  2. Mark Burgin
    Pages 11-40
  3. Mark Burgin
    Pages 41-69
  4. Mark Burgin
    Pages 71-97
  5. Mark Burgin
    Pages 99-135
  6. Mark Burgin
    Pages 137-141
  7. Back Matter
    Pages 143-160

About this book


“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.


differentiation extrafuntion hypernumber integration topology

Authors and affiliations

  • Mark Burgin
    • 1
  1. 1.Dept. MathematicsUniversity of California, Los AngelesLos AngelesUSA

Bibliographic information

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