Sokhotski–Plemelj 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

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

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:

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

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

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

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

${\displaystyle \lim _{\varepsilon \to 0^{+}}{\frac {1}{x\pm i\varepsilon }}=\mp i\pi \delta (x)+{\mathcal {P}}{{\Big (}{\frac {1}{x}}{\Big )}}.}$

where ${\displaystyle \delta (x)}$ is the Dirac delta function. This should be interpreted as an integral equality, 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 ${\displaystyle a<0<b}$. Then

${\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{a}^{b}{\frac {f(x)}{x\pm i\varepsilon }}\,dx=\mp i\pi f(0)+{\mathcal {P}}\int _{a}^{b}{\frac {f(x)}{x}}\,dx,}$

where ${\displaystyle {\mathcal {P}}}$ denotes the Cauchy principal value. (Note that this version makes no use of analyticity.)

A simple proof is as follows.

${\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{a}^{b}{\frac {f(x)}{x\pm i\varepsilon }}\,dx=\mp i\pi \lim _{\varepsilon \to 0^{+}}\int _{a}^{b}{\frac {\varepsilon }{\pi (x^{2}+\varepsilon ^{2})}}f(x)\,dx+\lim _{\varepsilon \to 0^{+}}\int _{a}^{b}{\frac {x^{2}}{x^{2}+\varepsilon ^{2}}}\,{\frac {f(x)}{x}}\,dx.}$

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 ∓iπ 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.

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

${\displaystyle \int _{-\infty }^{\infty }dE\,\int _{0}^{\infty }dt\,f(E)\exp(-iEt)}$

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 term to -iEt in the exponential, and then taking that to zero, i.e.:

${\displaystyle \lim _{\varepsilon \to 0^{+}}\int _{-\infty }^{\infty }dE\,\int _{0}^{\infty }dt\,f(E)\exp(-iEt-\varepsilon t)=-i\lim _{\varepsilon \to 0^{+}}\int _{-\infty }^{\infty }{\frac {f(E)}{E-i\varepsilon }}\,dE=\pi f(0)-i{\mathcal {P}}\int _{-\infty }^{\infty }{\frac {f(E)}{E}}\,dE,}$

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

• 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

• 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.{{cite book}}: CS1 maint: location missing publisher (link)