Cauchy principal value

Definition

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

• the finite number

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

• the finite number

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.