Improper Integral - Cauchy Principal Value

Cauchy Principal Value

Consider the difference in values of two limits:

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

\int_{-1}^1\frac{\mathrm{d}x}{x}{\ }
\left(\mbox{which}\ \mbox{gives}\ -\infty+\infty\right).

Similarly, we have

but

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

\int_{-\infty}^\infty\frac{2x\,\mathrm{d}x}{x^2+1}{\ }
\left(\mbox{which}\ \mbox{gives}\ -\infty+\infty\right).

All of the above limits are cases of the indeterminate form ∞ − ∞.

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

Read more about this topic:  Improper Integral

Famous quotes containing the word principal:

    Heaven has a Sea of Glass on which angels go sliding every afternoon. There are many golden streets, but the principal thoroughfares are Amen Street and Hallelujah Avenue, which intersect in front of the Throne. These streets play tunes when walked on, and all shoes have songs in them.
    —For the State of Florida, U.S. public relief program (1935-1943)