Definition:Cauchy Principal Value

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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:

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

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

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

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 $\phi: \left[{-R \,.\,.\, R}\right] \to \C$, 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:

$\displaystyle \operatorname {PV} \int_C f \left({z}\right) \rd z = \displaystyle \operatorname {PV} \int_{\phi \left({-\infty}\right)}^{\phi \left({+\infty}\right)} f \left({z}\right) \rd z := \lim_{R \mathop \to +\infty} \int_{-R}^R f \left({\phi \left({t}\right)}\right)\phi' \left({t}\right) \rd t$

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


Source of Name

This entry was named for Augustin Louis Cauchy.


Sources