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.

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