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.
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
of the integral: Subsequent generalizations relax the smoothness requirements on curve C and the function φ.
One may take the difference of these two equalities to obtain These formulae should be interpreted as integral equalities, 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
This expression, as written, is undefined (since the time integral does not converge), so it is typically modified by adding a negative real term to -iEt in the exponential, and then taking that to zero, i.e.: where the latter step uses the real version of the theorem.
In theoretical quantum optics, the derivation of a master equation in Lindblad form often requires the following integral function,[1] which is a direct consequence of the Sokhotski–Plemelj theorem and is often called the Heitler-function: