# Definition:Cauchy Principal Value/Complex Integral

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## Definition

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.

## 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.): $\S 7$