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:

$\mathrm{P}\mathrm{V}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(x)dx=\underset{r\to \mathrm{\infty }}{lim}\underset{-r}{\overset{r}{\int }}f(x)dx$

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

$\mathrm{P}\mathrm{V}\underset{a}{\overset{b}{\int }}f(x)dx=\underset{\epsilon \to 0^{+}}{lim}\underset{a}{\overset{c-\epsilon }{\int }}f(x)dx+\underset{c+\epsilon }{\overset{b}{\int }}f(x)dx$