Definition:Cauchy Principal Value/Contour Integral

Definition
Let $C$ be a contour defined by a directed smooth curve.

Let $C$ be parameterized by the smooth path $\phi: \closedint {-R} R \to \C$, where $R > 0$.

Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$.

Then the Cauchy principal value of $\ds \int f$ is defined as:


 * $\PV_C \map f z \rd z = \PV_{\map \phi {-\infty} }^{\map \phi {+\infty} } \map f z \rd z := \lim_{R \mathop \to +\infty} \int_{-R}^R \map f {\map \phi t} \map {\phi'} t \rd t$

where $\ds \int_{-R}^R \map f {\map \phi t} \map {\phi'} t \rd t$ is a complex Riemann integral defining a contour integral.