The Sokhotski–Plemelj theorem (Polish spelling is ''Sochocki'') is a
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
, which helps in evaluating certain integrals. The real-line version of it (
see below) is often used in physics, although rarely referred to by name. The theorem is named after
Julian Sochocki, who proved it in 1868, and
Josip Plemelj
Josip Plemelj (December 11, 1873 – May 22, 1967) was a Slovenes, Slovene mathematician, whose main contributions were to the theory of analytic functions and the application of integral equations to potential theory. He was the first chancel ...
, who rediscovered it as a main ingredient of his solution of the
Riemann–Hilbert problem in 1908.
Statement of the theorem
Let ''C'' be a smooth
closed simple curve in the plane, and
an
analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
on ''C''. Note that the
Cauchy-type integral
:
cannot be evaluated for any ''z'' on the curve ''C''. However, on the interior and exterior of the curve, the integral produces analytic functions, which will be denoted
inside ''C'' and
outside. The Sokhotski–Plemelj formulas relate the limiting boundary values of these two analytic functions at a point ''z'' on ''C'' and the
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 ...
of the integral:
:
:
Subsequent generalizations relax the smoothness requirements on curve ''C'' and the function ''φ''.
Version for the real line
Especially important is the version for integrals over the real line.
:
where
is the
Dirac delta function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
where
denotes the
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 ...
. One may take the difference of these two equalities to obtain
:
These formulae should be interpreted as integral equalities, as follows: Let be a
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued function which is defined and continuous on the real line, and let and be real constants with
. Then
:
and
:
Note that this version makes no use of analyticity.
Proof of the real version
A simple proof is as follows.
:
For the first term, is a
nascent delta function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
, and therefore approaches a
Dirac delta function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
in the limit. Therefore, the first term equals ∓''i'' ''f''(0).
For the second term, the factor approaches 1 for , ''x'', ≫ ''ε'', approaches 0 for , ''x'', ≪ ε, and is exactly symmetric about 0. Therefore, in the limit, it turns the integral into a
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 ...
integral.
Physics application
In
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
and
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, one often has to evaluate integrals of the form
:
where ''E'' is some energy and ''t'' is time. This expression, as written, is undefined (since the time integral does not converge), so it is typically modified by adding a negative real term to ''-iEt'' in the exponential, and then taking that to zero, i.e.:
:
where the latter step uses the real version of the theorem.
Heitler function
In
theoretical quantum optics, the derivation of a
master equation in Lindblad form often requires the following integral function,
which is a direct consequence of the Sokhotski–Plemelj theorem and is often called the Heitler-function:
:
See also
*
Singular integral operators on closed curves (account of the Sokhotski–Plemelj theorem for the unit circle and a closed Jordan curve)
*
Kramers–Kronig relations
The Kramers–Kronig relations, sometimes abbreviated as KK relations, are bidirectional mathematics, mathematical relations, connecting the real number, real and imaginary number, imaginary parts of any complex analysis, complex function that is a ...
*
Hilbert transform
In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, of a real variable and produces another function of a real variable . The Hilbert transform is given by the Cauchy principal value ...
References
Literature
* Chapter 3.1.
* Appendix A, equation (A.19).
*
*
*
*
* Blanchard, Bruening: Mathematical Methods in Physics (Birkhauser 2003), Example 3.3.1 4
*
{{DEFAULTSORT:Sokhotski-Plemelj theorem
Theorems in complex analysis