Sokhotski–Plemelj theorem


The Sokhotski–Plemelj theorem is a theorem in complex analysis, which helps in evaluating certain integrals. The real-line version of it 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, 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 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 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.
Let be a complex-valued function which is defined and continuous on the real line, and let and be real constants with. Then
where denotes the Cauchy principal value.

Proof of the real version

A simple proof is as follows.
For the first term, we note that is a nascent delta function, and therefore approaches a Dirac delta function in the limit. Therefore, the first term equals ∓i f.
For the second term, we note that 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 integral.
For simple proof of the complex version of the formula and version for polydomains see:

Physics application

In quantum mechanics and quantum field theory, one often has to evaluate integrals of the form
where E is some energy and t is time. This expression, as written, is undefined, so it is typically modified by adding a negative real coefficient to t in the exponential, and then taking that to zero, i.e.:
where the latter step uses the real version of the theorem.