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.


This article is complete as far as it goes, but it could do with expansion.
In particular: Needs to be include the situation where there is an undefined point at some $b \in \R$ where $\map f b$ is undefined, for example $\ds \int_{-\infty}^x \dfrac {e^u} u \rd u$ at $u = 0$
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Real Integral

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


Then the Cauchy principal value of $\ds \int f$ 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 $\ds \int_{-R}^R \map f t \rd t$ is a Riemann integral.


Complex Integral

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


Then the Cauchy principal value of $\ds \int f$ 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 $\ds \int_{-R}^R \map f t \rd t$ 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 $\ds \int f$ 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.


Also denoted as

Variants of the notation $\PV$ for the Cauchy principal value can often be seen, most of which use the letters $\text{PV}$, such as:

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

and so on.


Source of Name

This entry was named for Augustin Louis Cauchy.


Technical Note

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

This command is specific to $\mathsf{Pr} \infty \mathsf{fWiki}$.


Sources


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