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Singular Integrals and Fractal Calculus

  • Alexander I. SaichevEmail author
  • Wojbor Woyczynski
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

This chapter is devoted to integrals similar to the familiar divergent Cauchy integral
$$ \smallint \frac{\varphi (s)}{s - x}\mathop {ds}\limits^{{}} $$
(1)
Such integrals are often encountered in physical applications. If the function \( \varphi (s) \) does not vanish at \( s = x \) then the integrand in (1) has a nonintegrable singularity at that point. In practice physicists, using their intuition as a guide, often assign certain finite values to these integrals anyway. Then, it is a mathematician’s job to justify rigorously these “renormalizations of infinities”, translate additional physical requirements into mathematical terms and point out how different assumptions lead to different values of integral (1).

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Mathematical DepartmentState University of Nizhny NovgorodNizhny NovgorodRussia
  2. 2.Math, Applied Math, & Stat, Yost 229Case Western Reserve UniversityClevelandUSA

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