Definition:Cauchy Principal Value

Definition
The Cauchy Principal Value is an extension of the concept of an improper integral when the latter might not exist.

Real Integrals
Let $f: \R \to \R$ be a real function which is piecewise continuous everywhere.

Then the Cauchy Principal Value of $\displaystyle \int f$ is defined as:


 * $\PV_{-\infty}^{+\infty} \map f t \rd t := \lim_{R \mathop \to +\infty} \int_{-R}^R \map f t \rd t$

where $\displaystyle \int_{-R}^R \map f t \rd t$ is a Riemann integral.

Complex Integrals
Let $f: \R \to \C$ be a bounded complex function.

Then the Cauchy Principal Value of $\displaystyle \int f$ is defined as:


 * $\PV_{-\infty}^{+\infty} \map f t \rd t := \lim_{R \mathop \to +\infty} \int_{-R}^R \map f t \rd t$

where $\displaystyle \int_{-R}^R f \left({t}\right) \rd t$ is a complex Riemann integral.

Contour Integrals
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 $\displaystyle \int f$ 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 $\displaystyle \int_{-R}^R \map f {\map \phi t} \map {\phi'} t \rd t$ is a Complex Riemann Integral defining a Contour Integral.

Also denoted as
Variants of the letters $\text P$ and $\text V$ can often be seen, such as:


 * $\displaystyle \operatorname {P.V.} \int$


 * $\displaystyle \operatorname {p.v.} \int$


 * $\displaystyle PV \int$

and so on.