Cauchy principal value Guide, Meaning , Facts, Information and Description
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 "−").
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.
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