The Sokhotski–Plemelj theorem (Polish spelling is Sochocki; in English there are many different spellings) is a theorem in complex analysis, which helps in evaluating certain integrals. The realline 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 RiemannHilbert problem in 1908. Statement of the theorem Let C be a smooth closed simple curve in the plane, and φ an analytic function on C. Then the Cauchytype integral \( \int_C\frac{\phi(\zeta)d\zeta}{\zetaz}, \) defines two analytic functions, φ_{i} inside C and φ_{e} outside. Sokhotski–Plemelj formulas relate the boundary values of these two analytic functions at a point z on C and the principal value \( \mathcal{P} \) of the integral: \( \phi_i(z)=\frac{1}{2\pi i}\mathcal{P}\int_C\frac{\phi(\zeta) d\zeta}{\zetaz}+\frac{1}{2}\phi(z), \, \) \( \phi_e(z)=\frac{1}{2\pi i}\mathcal{P}\int_C\frac{\phi(\zeta) d\zeta}{\zetaz}\frac{1}{2}\phi(z). \, \) Subsequent generalizations relaxed the smoothness requirements on curve C and the function φ. Especially important is the version for integrals over the real line. Let ƒ be a complexvalued function which is defined and continuous on the real line, and let a and b be real constants with a < 0 < b. Then \( \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{f(x)}{x\pm i \varepsilon}\,dx = \mp i \pi f(0) + \mathcal{P}\int_a^b \frac{f(x)}{x}\, dx, \) where \( \mathcal{P} \) denotes the Cauchy principal value. A simple proof is as follows. \( \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{f(x)}{x\pm i \varepsilon}\,dx = \mp i \pi \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{\varepsilon}{\pi(x^2+\varepsilon^2)}f(x)\,dx + \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{x^2}{x^2+\varepsilon^2} \, \frac{f(x)}{x}\, dx. \) For the first term, we note that ε⁄π(x^{2} + ε^{2}) 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 x^{2}⁄(x^{2} + ε^{2}) 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. In quantum mechanics and quantum field theory, one often has to evaluate integrals of the form \( \int_{\infty}^\infty dE\, \int_0^\infty dt\, f(E)\exp(iEt) \) 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.: \( \lim_{\varepsilon\rightarrow 0^+} \int_{\infty}^\infty dE\, \int_0^\infty dt\, f(E)\exp(iEt\varepsilon t) \) \( = i \lim_{\varepsilon\rightarrow 0^+} \int_{\infty}^\infty \frac{f(E)}{Ei\varepsilon}\,dE = \pi f(0)i \mathcal{P}\int_{\infty}^{\infty}\frac{f(E)}{E}\,dE, \) where the latter step uses this theorem. Weinberg, Steven (1995). The Quantum Theory of Fields, Volume 1: Foundations. Cambridge Univ. Press. ISBN 0521550017. Chapter 3.1. Retrieved from "http://en.wikipedia.org/"


