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2014-01-11T17:48:06Z

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{{distinguish|Casorati–Sokhotski–Weierstrass 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 ([[#Version for the real line|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.

== Statement of the theorem ==

Let ''C'' be a smooth closed simple curve in the plane, and ''φ'' an analytic function on ''C''.
Then the '''Cauchy-type integral'''

:<math> \frac{1}{2\pi i} \int_C\frac{\phi(\zeta)d\zeta}{\zeta-z}, </math>

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 [[Cauchy principal value]] <math>\mathcal{P}</math> of the integral:

:<math> \phi_i(z)=\frac{1}{2\pi i}\mathcal{P}\int_C\frac{\phi(\zeta) d\zeta}{\zeta-z}+\frac{1}{2}\phi(z), \, </math>

:<math> \phi_e(z)=\frac{1}{2\pi i}\mathcal{P}\int_C\frac{\phi(\zeta) d\zeta}{\zeta-z}-\frac{1}{2}\phi(z). \, </math>

Subsequent generalizations relaxed the smoothness requirements on curve ''C'' and the function ''φ''.

==Version for the real line==

Especially important is the version for integrals over the real line.

Let ''&fnof;'' be a [[complex number|complex]]-valued function which is defined and continuous on the real line, and let ''a'' and ''b'' be real constants with ''a'' < 0 < ''b''. Then

:<math>\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,</math>

where <math>\mathcal{P}</math> denotes the [[Cauchy principal value]].

== Proof of the real version ==

A simple proof is as follows.

:<math>
\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.</math>

For the first term, we note that {{frac|''ε''|{{pi}}(''x''2 + ''ε''2)}} is a [[nascent delta function]], and therefore approaches a [[Dirac delta function]] in the limit. Therefore, the first term equals ∓''i''{{pi}} ''f''(0).

For the second term, we note that the factor {{fraction|''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.

==Physics application ==

In [[quantum mechanics]] and [[quantum field theory]], one often has to evaluate integrals of the form

:<math>\int_{-\infty}^\infty dE\, \int_0^\infty dt\, f(E)\exp(-iEt)</math>

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.:

:<math>\lim_{\varepsilon\rightarrow 0^+} \int_{-\infty}^\infty dE\, \int_0^\infty dt\, f(E)\exp(-iEt-\varepsilon t)</math>

::<math>= -i \lim_{\varepsilon\rightarrow 0^+} \int_{-\infty}^\infty \frac{f(E)}{E-i\varepsilon}\,dE = \pi f(0)-i \mathcal{P}\int_{-\infty}^{\infty}\frac{f(E)}{E}\,dE,</math>
where the latter step uses this theorem.

==See also==
*[[Singular_integral_operators_on_closed_curves#Plemelj-Sokhotski_relation|Singular integral operators on closed curves]] (account of the Sokhotski–Plemelj theorem for the unit circle and a closed Jordan curve)
*[[Kramers-Kronig relations]]
*[[Hilbert transform]]

== References ==
{{reflist}}
* {{cite book | authorlink=Steven Weinberg | author=Weinberg, Steven | title=The Quantum Theory of Fields, Volume 1: Foundations | publisher=Cambridge Univ. Press | year=1995 | isbn=0-521-55001-7}} Chapter 3.1.
* {{cite book | author=Merzbacher, Eugen | title=Quantum Mechanics | publisher=Wiley, John & Sons, Inc. | year=1998 | isbn=0-471-88702-1}} Appendix A, equation (A.19).
* {{cite book | author=Henrici, Peter | title=Applied and Computational Complex Analysis, vol. 3|publisher=Willey, John & Sons, Inc.| year=1986 }}
* {{cite book | author=Plemelj, Josip | title=Problems in the sense of Riemann and Klein | publisher=Interscience Publishers|place=New York|year= 1964}}
*{{citation|last=Gakhov|first= F. D.|title=Boundary value problems. Reprint of the 1966 translation|publisher= Dover Publications|year=1990|isbn=0-486-66275-6}}
* {{cite book | author=Muskhelishvili, N. I.|title=Singular integral equations, boundary problems of function theory and their application to mathematical physics|
publisher=Dept. of Supply and Development, Aeronautical Research Laboratories|place=Melbourne|year=1949}}
* Blanchard, Bruening: Mathematical Methods in Physics (Birkhauser 2003), Example 3.3.1 4

{{DEFAULTSORT:Sokhotski-Plemelj theorem}}
[[Category:Theorems in complex analysis]]

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