Cauchy principal value

In mathematics, the Cauchy principal value of certain improper integrals is defined as either

Related Topics:
Mathematics - Improper integral

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• the finite number

::lim_{arepsilon ightarrow 0+} left

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:where b is a point at which the behavior of the function f is such that

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::int_a^b f(x),dx=pminfty

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:for any a < b and

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::int_b^c f(x),dx=mpinfty

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:for any c > b (one sign is "+" and the other is "−").

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or

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• the finite number

::lim_{a ightarrowinfty}int_{-a}^a f(x),dx

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:where

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::int_{-infty}^0 f(x),dx=pminfty

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:and

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::int_0^infty f(x),dx=mpinfty

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:(again, one sign is "+" and the other is "−").

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In some cases it is necessary to deal simultaneously with singularities both at a finite number b and at infinity. This is usually done by a limit of the form

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::lim_{arepsilon ightarrow 0+}int_{b-rac{1}{arepsilon}}^{b-arepsilon} f(x),dx+int_{b+arepsilon}^{b+rac{1}{arepsilon}}f(x),dx.

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Cauchy principal value: Nomenclature ►