The Sokhotski–Plemelj theorem (Polish spelling is Sochocki) is a theorem in complex analysis, 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, who rediscovered it as a main ingredient of his solution of the Riemann-Hilbert problem in 1908.
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Statement of the theorem
Let C be a smooth closed simple curve in the plane, and
defines two analytic functions of z,
Subsequent generalizations relaxed 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 f be a complex-valued function which is defined and continuous on the real line, and let a and b be real constants with a < 0 < b. Then
where
Proof of the real version
A simple proof is as follows.
For the first term, we note that ε⁄π(x2 + ε2) is a nascent delta function, and therefore approaches a Dirac delta function in the limit. Therefore, the first term equals ∓iπ f(0).
For the second term, we note that the factor x2⁄(x2 + ε2) 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.
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 (since the time integral does not converge), 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.