Cauchy Principal Value - Examples

Examples

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).

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