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
\varphi
\phi(z)=
| 1 | |
| 2\pii |
| \int | ||||
|
,
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
\phii
\phie
l{P}
\limw\phii(w)=
| 1 | |
| 2\pii |
| l{P}\int | ||||
|
+
| 1 | |
| 2 |
\varphi(z),
\limw\phie(w)=
| 1 | |
| 2\pii |
| l{P}\int | - | ||||
|
| 1 | |
| 2 |
\varphi(z).
Subsequent generalizations relax the smoothness requirements on curve C and the function φ.
See also: Kramers–Kronig relations. Especially important is the version for integrals over the real line.
| \lim | |
| \varepsilon\to0+ |
| 1 | |
| x\pmi\varepsilon |
=\mpi\pi\delta(x)+{l{P}}{(
| 1 | |
| x |
)}.
where
\delta(x)
l{P}
| \lim | |
| \varepsilon\to0+ |
\left[
| 1 | |
| x-i\varepsilon |
-
| 1 | |
| x+i\varepsilon |
\right]=2\pii\delta(x).
These formulae should be interpreted as integral equalities, as follows: Let be a complex-valued function which is defined and continuous on the real line, and let and be real constants with
a<0<b
| \lim | |
| \varepsilon\to0+ |
| b | |
| \int | |
| a |
| f(x) | |
| x\pmi\varepsilon |
dx=\mpi\pif(0)+
| b | |
| l{P}\int | |
| a |
| f(x) | |
| x |
dx
and
| \lim | |
| \varepsilon\to0+ |
| b | |
| \int | |
| a |
\left[
| f(x) | |
| x-i\varepsilon |
-
| f(x) | |
| x+i\varepsilon |
\right]dx=2\piif(0)
Note that this version makes no use of analyticity.
A simple proof is as follows.
| \lim | |
| \varepsilon\to0+ |
| b | |
| \int | |
| a |
| f(x) | |
| x\pmi\varepsilon |
dx=\mpi\pi
| \lim | |
| \varepsilon\to0+ |
| b | |
| \int | |
| a |
| \varepsilon | |
| \pi(x2+\varepsilon2) |
f(x)dx+
| \lim | |
| \varepsilon\to0+ |
| b | |
| \int | |
| a |
| x2 | |
| x2+\varepsilon2 |
| f(x) | |
| x |
dx.
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 ∓i 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.
In quantum mechanics and quantum field theory, one often has to evaluate integrals of the form
| infty | |
| \int | |
| -infty |
dE
| infty | |
| \int | |
| 0 |
dtf(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 term to -iEt in the exponential, and then taking that to zero, i.e.:
| \lim | |
| \varepsilon\to0+ |
| infty | |
| \int | |
| -infty |
dE
| infty | |
| \int | |
| 0 |
dtf(E)\exp(-iEt-\varepsilont)=-i
| \lim | |
| \varepsilon\to0+ |
| infty | |
| \int | |
| -infty |
| f(E) | |
| E-i\varepsilon |
dE=\pif(0)-i
| infty | |
| l{P}\int | |
| -infty |
| f(E) | |
| E |
dE,
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:
| infty | |
| \int | |
| 0 |
d\tau\exp(-i(\omega\pm\nu)\tau)=\pi\delta(\omega\pm\nu)-il{P}(
| 1 | |
| \omega\pm\nu |
)
. Steven Weinberg . Weinberg, Steven . The Quantum Theory of Fields, Volume 1: Foundations . Cambridge Univ. Press . 1995 . 0-521-55001-7 . registration . Chapter 3.1.