Definition:Cauchy Principal Value/Real Integral

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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 .