Definition:Cauchy Principal Value/Real Integral
Definition
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.
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
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- 2004: James Ward Brown and Ruel V. Churchill: Complex Variables and Applications (7th ed.): $\S 7$