Finite Part (other)

Finite Part (other)
Finite part may refer to: *Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Formulation Depending on the type of singularity in the integrand ... * Hadamard finite part {{mathdab ...
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Cauchy Principal Value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Formulation Depending on the type of singularity in the integrand , the Cauchy principal value is defined according to the following rules: In some cases it is necessary to deal simultaneously with singularities both at a finite number and at infinity. This is usually done by a limit of the form \lim_\, \lim_ \,\left ,\int_^ f(x)\,\mathrmx \,~ + ~ \int_^ f(x)\,\mathrmx \,\right In those cases where the integral may be split into two independent, finite limits, \lim_ \, \left, \,\int_a^ f(x)\,\mathrmx \,\\; < \;\infty and

\lim_\;\left, \,\int_^c f(x)\,\mathrmx \,\ \; < \; \infty ,

then the function is integrable in the ordinary sense. The result of the procedure for principal value is the same as the ordinary integral; since it no longer matches the definition, ...
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