# Cauchy principal value

## Definition

In mathematics, the **Cauchy principal value** of certain improper integrals is defined as either

where *b* is a point at which the behavior of the function *f* is such that

for any *a* < *b* and

for any *c* > *b* (one sign is "+" and the other is "&minus").

or

where

and

(again, one sign is "+" and the other is "−").

## Examples

Consider the difference in values of two limits:

The former is the **Cauchy principal value** of the otherwise ill-defined expression

Similarly, we have

but

The former is the principal value of the otherwise ill-defined expression

These pathologies do not afflict Lebesgue-integrable functions, that is, functions the integrals of whose absolute values are finite.