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:

    I note what you say of the late disturbances in your College. These dissensions are a great affliction on the American schools, and a principal impediment to education in this country.
    Thomas Jefferson (1743–1826)