Definition:Cauchy Principal Value/Complex Integral

Definition

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.

Also denoted as

Variants of the notation $\PV$ for the Cauchy principal value can often be seen, such as:

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

and so on.

Source of Name

This entry was named for Augustin Louis Cauchy.

Technical Note

The $\LaTeX$ code for $\PV$ is \PV .