Singular Integral Equations

Abstract

Let the function f be defined on I=[a,b] and, possibly, be singular at an interior point c∈(a,b). Recall that the improper integral was defined by

$$\int\limits_{a}^{b} {f\left( x \right)} dx: = \mathop{{\lim }}\limits_{{\mathop{{{{\varepsilon }_{1}} \to 0}}\limits_{{{{\varepsilon }_{1}} > 0}} }} \int\limits_{a}^{{c - {{\varepsilon }_{1}}}} {f\left( x \right)} dx + \mathop{{\lim }}\limits_{{\mathop{{{{\varepsilon }_{2}} \to 0}}\limits_{{{{\varepsilon }_{2}} > 0}} }} \int\limits_{{c + {{\varepsilon }_{2}}}}^{b} {f\left( x \right)dx,} $$

if both limits exist (cf. §6.1.3). By Remark 6.1.2a, the improper integral exists for f (x): = |x-c|^s with s>-1. For $f(x): = \frac{1}{{x - c}}({\text{i}}{\text{.e}}{\text{.,s = - 1}})$ (i.e, s=-1) one obtains

$$\int\limits_{a}^{{c - {{\varepsilon }_{1}}}} {\frac{1}{{x - c}}dx} + \int\limits_{{c + {{\varepsilon }_{2}}}}^{b} {\frac{1}{{x - c}}dx = \log \frac{{b - c}}{{c - a}} + \log \frac{{{{\varepsilon }_{1}}}}{{{{\varepsilon }_{2}}}}.} $$

(7.1.1)