SINGULAR DEFINITE INTEGRALS

Dennis M. Cates ()

Chapter Chapter 25 in Cauchy's Calcul Infinitésimal, 2019, pp 133-136 from Springer

Abstract: Abstract Consider that an integral relative to x, and in which the function under the $$\int $$ sign is denoted by f(x), is taken between two limits infinitely close to a definite particular value a attributed to the variable $$x. \ $$ If this value a is a finite quantity, and if the function f(x) remains finite and continuous in the neighborhood of $$x=a, $$ then, by virtue of formula ( 19 ) (twenty-second lecture), the proposed integral will be essentially null. But, it can obtain a finite value different from zero or even an infinite value, ifImproper integral we have $$\begin{aligned} a=\displaystyle \frac{\pm }{\infty } \ \ \ \ \ \ \ \ \text {or else} \ \ \ \ \ \ \ \ f(a)=\pm \infty . \end{aligned}$$ In this latter case, the integral in question will become what we will call a singular definite integral.Singular definite integral It will ordinarily be easy to calculate its value with the help of formulas ( 15 ) and ( 16 ) of the twenty-third lecture, as we shall see.

Date: 2019
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DOI: 10.1007/978-3-030-11036-9_25

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