Indented real-axis contours compute principal values with half-residue corrections

Let $R=p\mathrm{/}q$, where $p,q\in \mathbb{C}[z]$ are nonzero, be a rational function with $\mathrm{deg}q\ge \mathrm{deg}p+2$, and assume that its real poles $a_1<\cdots <a_s$ are simple. For $T$ large enough to contain all these poles and for pairwise disjoint symmetric deletion radii ${\epsilon }_j>0$, put

$I(T,\mathit{\epsilon }):={\int }_{[-T,T]\setminus {\bigcup }_{j=1}^s(a_j-{\epsilon }_j,a_j+{\epsilon }_j)}R(x)dx,$

where the integral is the sum over the remaining compact intervals. In this statement,

$\mathrm{PV}\text{\hspace{0.17em}⁣}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}R(x)dx:={\lim }_{\begin{array}{c}T\to \mathrm{\infty }\\ {\epsilon }_1,\dots ,{\epsilon }_s\downarrow 0\end{array}}I(T,\mathit{\epsilon })\mathrm{.}$

When $s=0$, the deleted union is empty and only $T\to \mathrm{\infty }$ remains. Indent every real pole above the real axis and close the contour in the upper half-plane. Then this principal value exists and

$\mathrm{PV}\text{\hspace{0.17em}⁣}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}R(x)dx=2\pi i{\sum }_{\mathrm{\Im }a>0}\mathrm{Res}(R,a)+i\pi {\sum }_j\mathrm{Res}(R,a_j)\mathrm{.}$

The correction term is the sum of the usual positive half-residues.