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).

Read more about this topic:  Cauchy Principal Value

Famous quotes containing the word examples:

    Histories are more full of examples of the fidelity of dogs than of friends.
    Alexander Pope (1688–1744)

    In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.
    Michel de Montaigne (1533–1592)

    No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.
    André Breton (1896–1966)