# Singular Integrals and Fractal Calculus

Chapter

## Abstract

This chapter is devoted to integrals similar to the familiar divergent Cauchy integral 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).

$$ \smallint \frac{\varphi (s)}{s - x}\mathop {ds}\limits^{{}} $$

(1)

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