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:


 * $\displaystyle \operatorname{PV}\int_{-\infty}^{+\infty} f \left({t}\right) \ \mathrm d t := \lim_{R \to +\infty}\int_{-R}^R f \left({t}\right) \ \mathrm d t$

where $\displaystyle \int_{-R}^R f \left({t}\right) \ \mathrm d t$ 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:


 * $\displaystyle \operatorname{PV}\int_{-\infty}^{+\infty} f \left({t}\right) \ \mathrm d t := \lim_{R \to +\infty}\int_{-R}^R f \left({t}\right) \ \mathrm d t$

where $\displaystyle \int_{-R}^R f \left({t}\right) \ \mathrm d t$ 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 $\phi: \left[{-R\,.\,.\,R}\right] \to \C$, where $R > 0$.

Let $f: \operatorname{Im} \left({C}\right) \to \C$ be a continuous complex function, where $\operatorname{Im} \left({C}\right)$ denotes the image of $C$.

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


 * $\displaystyle \operatorname{PV}\int_{C} f \left({z}\right) \ \mathrm d z = \displaystyle \operatorname{PV}\int_{\phi\left({-\infty}\right)}^{\phi\left({+\infty}\right)} f \left({z}\right) \ \mathrm d z := \lim_{R \to +\infty}\int_{-R}^R f \left({\phi\left({t}\right)}\right)\phi'\left({t}\right) \ \mathrm d t$

where $\displaystyle \int_{-R}^R f \left({\phi\left({t}\right)}\right)\phi'\left({t}\right) \ \mathrm d t$ 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.