Sokhotski–Plemelj 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 (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 $\varphi$ an analytic function on C. Then the Cauchy-type integral

{\frac {1}{2\pi i}}\int _{C}{\frac {\varphi (\zeta )\,d\zeta }{\zeta -z}},

defines two analytic functions of z, $\phi _{i}$ inside C and $\phi _{e}$ 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 ${\mathcal {P}}$ of the integral:

\phi _{i}(z)={\frac {1}{2\pi i}}{\mathcal {P}}\int _{C}{\frac {\varphi (\zeta )\,d\zeta }{\zeta -z}}+{\frac {1}{2}}\varphi (z),\,

\phi _{e}(z)={\frac {1}{2\pi i}}{\mathcal {P}}\int _{C}{\frac {\varphi (\zeta )\,d\zeta }{\zeta -z}}-{\frac {1}{2}}\varphi (z).\,

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 $f$ be a 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

\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. (Note that this version makes no use of analyticity.)

Proof of the real version

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⁄_{$π$ (x² + ε²)} is a nascent delta function, and therefore approaches a Dirac delta function in the limit. Therefore, the first term equals ∓i $π$ f(0).

For the second term, we note that the factor ^x²⁄_{(x² + ε²)} 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

\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)}{E-i\varepsilon }}\,dE=\pi f(0)-i{\mathcal {P}}\int _{-\infty }^{\infty }{\frac {f(E)}{E}}\,dE,

where the latter step uses the real version of the theorem.

References

Weinberg, Steven (1995). The Quantum Theory of Fields, Volume 1: Foundations. Cambridge Univ. Press. ISBN 0-521-55001-7. Chapter 3.1.
Merzbacher, Eugen (1998). Quantum Mechanics. Wiley, John & Sons, Inc. ISBN 0-471-88702-1. Appendix A, equation (A.19).
Henrici, Peter (1986). Applied and Computational Complex Analysis, vol. 3. Willey, John & Sons, Inc.
Plemelj, Josip (1964). Problems in the sense of Riemann and Klein. New York: Interscience Publishers.
Gakhov, F. D. (1990), Boundary value problems. Reprint of the 1966 translation, Dover Publications, ISBN 0-486-66275-6
Muskhelishvili, N. I. (1949). Singular integral equations, boundary problems of function theory and their application to mathematical physics. Melbourne: Dept. of Supply and Development, Aeronautical Research Laboratories.
Blanchard, Bruening: Mathematical Methods in Physics (Birkhauser 2003), Example 3.3.1 4
Sokhotskii, Y. W. (1873). On definite integrals and functions used in series expansions. St. Petersburg.

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