The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals (Q2798698)

A new approach to the concept of a principal value for a (divergent) integral of the form NEWLINE\[NEWLINE\int_a^b\frac{f(x)\,dx}{(x-x_0)^n},\tag{\(\star\)}NEWLINE\]NEWLINE \(n\in\mathbb N\), \(a<x_0<b\), is offered. Assuming that \(f\) can be extended to an analytic function in a region \(R\supset[a,b]\) of the complex plane, the \textit{analytic principal value} of \((\star)\) is defined as follows. Take in \(R\setminus\{x_0\}\) any contour \(\gamma^+\) that starts at \(a\), ends at \(b\), and is contained in the upper half plane. Similarly, a contour \(\gamma^-\) is found in the lower half plane. Then set NEWLINE\[NEWLINE\int_a^b\frac{f(x)\,dx}{(x-x_0)^n}=\frac12\left(\oint_{\gamma^+}\frac{f(x)\,dx}{(x-x_0)^n}+\oint_{\gamma^-}\frac{f(x)\,dx}{(x-x_0)^n}\right).NEWLINE\]NEWLINE This definition does not depend on the choice of \(\gamma^+,\gamma^-\). The connections of the analytic principal value to some similar concepts, namely the ordinary (Cauchy) principal value and the so-called Finite Part Integral, are discussed.