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 Integral

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 Integral

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


Contour Integral

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 notation $\PV$ for the Cauchy principal value can often be seen, such as:

  • $\operatorname {P.V.} \displaystyle \int$
  • $\operatorname {p.v.} \displaystyle \int$
  • $PV \displaystyle \int$

and so on.


Source of Name

This entry was named for Augustin Louis Cauchy.


Technical Note

The $\LaTeX$ code for \(\PV\) is \PV .


Sources