Cauchy principal value

A limiting prescription for certain improper integrals that balances contributions by excluding a singularity symmetrically, or truncating infinite tails symmetrically, before taking a limit.

Improper integral: An integral defined through a limit when its interval is unbounded or its integrand is unbounded. The principal value modifies how an improper integral’s limiting process is taken.
Hilbert transform: An integral transform whose kernel is interpreted through a Cauchy principal value. Its singular kernel makes symmetric cancellation part of the transform’s definition.
Lebesgue integral: An integral defined using measure and approximation by simple functions. Its existence requires integrability, rather than cancellation imposed by symmetric cutoffs.
Limit (mathematics): A value that a function or sequence approaches as its input approaches a specified point or infinity. A principal value is defined by a particular arrangement of limits.
Symmetric limit: A limit formed by approaching a point from both sides with matched offsets. Matched offsets define the cancellation at an interior singularity.
Fourier transform: A transform that represents functions or distributions by their frequency components. Principal values arise in Fourier transforms of nonintegrable oscillatory kernels.
Riemann integral: An integral defined as the limit of sums over partitions of an interval. Ordinary integrability does not generally assign values to singular integrals this way.
Cauchy criterion: A condition for convergence stated in terms of terms or values becoming arbitrarily close to one another. It distinguishes genuine convergence from a finite value produced only by balanced truncation.
Hadamard finite part: A regularization that extracts a finite remainder after subtracting divergent terms from an integral. Unlike principal value, it can remove divergences that symmetric cancellation leaves behind.
Kramers–Kronig relations: Relations connecting the real and imaginary parts of causal response functions. Their integral formulas commonly use principal values at singular denominators.

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