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 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
- 2004: James Ward Brown and Ruel V. Churchill: Complex Variables and Applications (7th ed.): $\S 7$