Sokhotski–Plemelj theorem

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. 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. Kramers–Kronig relations Especially important is the version for integrals over the real line. where is the Dirac delta function. This should be interpreted as an integral equality, as follows. 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 . Then where denotes the Cauchy principal value. (Note that this version makes no use of analyticity.) 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 ∓ipi f(0). 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: 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.