Cauchy principal value

Definition

The Cauchy principal value is a particular way of interpreting improper definite integrals in terms of how we approach the limits. The Cauchy principal value for integrating functions along the real axis is defined as:

$\operatorname {PV} \int _{-\infty }^{\infty }f(x)\,dx=\lim _{r\to \infty }\int _{-r}^{r}f(x)\,dx$

Similarly, if a function $f$ on an interval $(a,b)$ has a singularity at a point $c\in (a,b)$, we define:

$\operatorname {PV} \int _{a}^{b}f(x)\,dx=\lim _{\varepsilon \to 0^{+}}\int _{a}^{c-\varepsilon }f(x)\,dx+\int _{c+\varepsilon }^{b}f(x)\,dx$