Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
The Cauchy principal value of a function which is integrable on the complement of one point is, if it exists, the limit of the integrals of the function over subsets in the complement of this point as these integration domains tend to that point symmetrically from all sides.
One also subsumes the case that the “point” is “at infinity”, hence that the function is integrable over every bounded domain. In this case the Cauchy principal value is the limit, if it exists, of the integrals of the function over bounded domains, as their bounds tend symmetrically to infinity.
The operation of sending a compactly supported smooth function (bump function) to Cauchy principal value of its pointwise product with a function that may be singular at the origin defines a distribution, usually denoted .
When the Cauchy principal value exists but the full integral does not (hence when the full integral “diverges”) one may think of the Cauchy principal value as “exracting a finite value from a diverging quantity”. This is similar to the intuition of the early days of renormalization in perturbative quantum field theory (Schwinger-Tomonaga-Feynman-Dyson), but one has to be careful not to carry this analogy too far.
One point where the Cauchy principal value really does play a key role in perturbative quantum field theory is in the computation of Green functions (propagators) for the Klein-Gordon operator and the Dirac operator. See remark below and see at Feynman propagator for more on this.
(Cauchy principal value of an integral over the real line)
Let be a function on the real line such that for every positive real number its restriction to is integrable. Then the Cauchy principal value of is, if it exists, the limit
(Cauchy principal value as distribution on the real line)
Let be a function on the real line such that for all bump functions the Cauchy principal value of the pointwise product function exists, in the sense of def. . Then this assignment
defines a distribution .
Let be an integrable function which is symmetric, in that for all . Then the principal value integral (def. ) of exists and is zero:
This is because, by the symmetry of and the skew-symmetry of , the two contributions to the integral are equal up to a sign:
The principal value distribution (def. ) solves the distributional equation
Since the delta distribution solves the equation
we have that more generally every linear combination of the form
for , is a distributional solution to .
The wave front set of these solutions is
The first statement is immediate from the definition: For any bump function we have that
Regarding the second statement: One computes that the Fourier transforms (with oscillation factor and normalization factor 1) of and are given by and , respectively. From this the statement immediately follows.
This follows by the characterization of extension of distributions to a point, see there at this prop. (Hörmander 90, thm. 3.2.4)
(integration against inverse variable with imaginary offset)
Write
for the distribution which is the limit in of the non-singular distributions which are given by the smooth functions as the positive real number tends to zero:
hence the distribution which sends to
(Cauchy principal value equals integration with imaginary offset plus delta distribution)
The Cauchy principal value distribution (def. ) is equal to the sum of the integration over with imaginary offset (def. ) and a delta distribution.
(Plemelj-Sochocki formula)
In particular, by prop. this means that solves the distributional equation
Using that
we have for every bump function
Since
it is plausible that , and similarly that . In detail:
and
where we used that the derivative of the arctan function is and that is proportional to the sign function.
(Fourier integral formula for step function)
The Heaviside distribution is equivalently the following Cauchy principal value:
where the limit is taken over sequences of positive real numbers tending to zero.
We may think of the integrand uniquely extended to a holomorphic function on the complex plane and consider computing the given real line integral for fixed as a contour integral in the complex plane.
If is positive, then the exponent
has negative real part for positive imaginary part of . This means that the line integral equals the complex contour integral over a contour closing in the upper half plane. Since has positive imaginary part by construction, this contour does encircle the pole of the integrand at . Hence by the Cauchy integral formula in the case one gets
Conversely, for the real part of the integrand decays as the negative imaginary part increases, and hence in this case the given line integral equals the contour integral for a contour closing in the lower half plane. Since the integrand has no pole in the lower half plane, in this case the Cauchy integral formula says that this integral is zero.
Conversely, by the Fourier inversion theorem, the Fourier transform of the Heaviside distribution is the Cauchy principal value as in prop. :
(relation to Fourier transform of Heaviside distribution / Schwinger parameterization)
Here the second equality is also known as complex Schwinger parameterization.
As generalized functions consider the limit with a decaying component:
Let be a non-degenerate real quadratic form analytically continued to a real quadratic form
Write for the determinant of
Write for the induced quadratic form on dual vector space. Notice that (and hence ) are assumed non-degenerate but need not necessarily be positive or negative definite.
(Fourier transform of principal value of power of quadratic form)
Let be any real number, and any complex number. Then the Fourier transform of distributions of is
where
deotes the Gamma function
denotes the modified Bessel function.
Notice that diverges for as (DLMF 10.30.2).
(Gel’fand-Shilov 66, III 2.8 (8) and (9), p 289)
Let be the dual Minkowski metric in dimension . Then
is the Feynman propagator for the Klein-Gordon equation on Minkowski spacetime. In this case prop. implies that its singular support is the light cone .
Let be a non-degenerate real quadratic form analytically continued to a real quadratic form
Write for the determinant of . Write for the number of negative eigenvalues.
Write for the induced quadratic form on dual vector space. Notice that (and hence ) are assumed non-degenerate but need not necessarily be positive or negative definite.
(Fourier transform of delta distribution applied to mass shell)
Let , then the Fourier transform of distributions of the delta distribution applied to the “mass shell” is
where denotes the modified Bessel function of order .
Notice that diverges for as (DLMF 10.30.2).
(Gel’fand-Shilov 66, III 2.11 (7), p 294)
Let be the dual Minkowski metric in dimension . Then
is the causal propagator for the Klein-Gordon equation on Minkowski spacetime. In this case prop. implies that its singular support is the light cone .
Named after Augustin Cauchy
Detailed discussion of relation to Bessel functions is in
References on homogeneous distributions
See also
Wikipedia, Cauchy principal value
Wikipedia, Hadamard principal value
Last revised on September 29, 2023 at 06:20:24. See the history of this page for a list of all contributions to it.