Sokhotski–Plemelj theorem

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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.

Contents 1 Statement of the theorem 2 Version for the real line 3 Proof of the real version 4 Physics application 4.1 Heitler function 5 See also 6 References 7 Literature Statement of the theorem Let C be a smooth closed simple curve in the plane, and {displaystyle varphi } an analytic function on C. Note that the Cauchy-type integral {displaystyle phi (z)={frac {1}{2pi i}}int _{C}{frac {varphi (zeta ),dzeta }{zeta -z}},} 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 {displaystyle phi _{i}} inside C and {displaystyle phi _{e}} 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 {displaystyle {mathcal {P}}} of the integral: {displaystyle lim _{wto z}phi _{i}(w)={frac {1}{2pi i}}{mathcal {P}}int _{C}{frac {varphi (zeta ),dzeta }{zeta -z}}+{frac {1}{2}}varphi (z),} {displaystyle lim _{wto z}phi _{e}(w)={frac {1}{2pi i}}{mathcal {P}}int _{C}{frac {varphi (zeta ),dzeta }{zeta -z}}-{frac {1}{2}}varphi (z).} Subsequent generalizations relax the smoothness requirements on curve C and the function φ.

Version for the real line See also: Kramers–Kronig relations 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 {displaystyle a<0

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