This article is about a method for assigning values to improper integrals. For the values of a complex function associated with a single branch, see Principal value. For the negative-power portion of a Laurent series, see Principal part.
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of a complex-valued function f(z); z = x + iy, with a pole on a contour C. Define C(ε) to be the same contour where the portion inside the disk of radius ε around the pole has been removed. Provided the function f(z) is integrable over C(ε) no matter how small ε becomes, then the Cauchy principal value is the limit:
In the case of Lebesgue-integrable functions, that is, functions which are integrable in absolute value, these definitions coincide with the standard definition of the integral.
If the function f(z) is meromorphic, the Sokhotski–Plemelj theorem relates the principal value of the integral over C with the mean-value of the integrals with the contour displaced slightly above and below, so that the residue theorem can be applied to those integrals.
Principal value integrals play a central role in the discussion of Hilbert transforms.
is a distribution. The map itself may sometimes be called the principal value (hence the notation p.v.). This distribution appears, for example, in the Fourier transform of the Heaviside step function.
Note that the proof needs merely to be continuously differentiable in a neighbourhood of and to be bounded towards infinity. The principal value therefore is defined on even weaker assumptions such as integrable with compact support and differentiable at 0.
The principal value is the inverse distribution of the function and is almost the only distribution with this property:
where is a constant and the Dirac distribution.
In a broader sense, the principal value can be defined for a wide class of singular integralkernels on the Euclidean space . If has an isolated singularity at the origin, but is an otherwise "nice" function, then the principal-value distribution is defined on compactly supported smooth functions by
Such a limit may not be well defined, or, being well-defined, it may not necessarily define a distribution. It is, however, well-defined if is a continuous homogeneous function of degree whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms.