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. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the non singular domain.

Formulation edit

Depending on the type of singularity in the integrand f, the Cauchy principal value is defined according to the following rules:

For a singularity at a finite number b
with   and where b is the difficult point, at which the behavior of the function f is such that
for any   and
for any   (See plus or minus for the precise use of notations ± and ∓.)
For a singularity at infinity ( )

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

In those cases where the integral may be split into two independent, finite limits,
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, it is technically not a "principal value". The Cauchy principal value can also be defined in terms of contour integrals of a complex-valued function   with   with a pole on a contour C. Define   to be that same contour, where the portion inside the disk of radius ε around the pole has been removed. Provided the function   is integrable over   no matter how small ε becomes, then the Cauchy principal value is the limit:[1]
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   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.[2]

Distribution theory edit

Let   be the set of bump functions, i.e., the space of smooth functions with compact support on the real line  . Then the map

defined via the Cauchy principal value as
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 sign function and the Heaviside step function.

Well-definedness as a Distribution edit

To prove the existence of the limit

for a Schwartz function  , first observe that   is continuous on   as
and hence
since   is continuous and L'Hopital's rule applies.

Therefore,   exists and by applying the mean value theorem to   we get:


And furthermore:


we note that the map

is bounded by the usual seminorms for Schwartz functions  . Therefore, this map defines, as it is obviously linear, a continuous functional on the Schwartz space and therefore a tempered distribution.

Note that the proof needs   merely to be continuously differentiable in a neighbourhood of 0 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.

More general definitions edit

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 integral kernels 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.

Examples edit

Consider the values of two limits:


This is the Cauchy principal value of the otherwise ill-defined expression




Similarly, we have


This is the principal value of the otherwise ill-defined expression


Notation edit

Different authors use different notations for the Cauchy principal value of a function  , among others:

as well as   P.V.,       and V.P.

See also edit

References edit

  1. ^ Kanwal, Ram P. (1996). Linear Integral Equations: Theory and technique (2nd ed.). Boston, MA: Birkhäuser. p. 191. ISBN 0-8176-3940-3 – via Google Books.
  2. ^ King, Frederick W. (2009). Hilbert Transforms. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-88762-5.