Definition:Cauchy Principal Value/Contour Integral

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Definition

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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