Sokhatsky-Weierstrass theorem

The Sokhatsky-Weierstrass theorem (also spelled Sokhotsky-Weierstrass theorem, and also called the Weierstrass theorem, although the latter term has several, more common, alternate meanings) is a theorem in complex analysis, which helps in evaluating certain Cauchy-type integrals, among many other applications. It is often used in physics, although rarely referred to by name. The theorem is named after Yulian Sokhotski and Karl Weierstrass.

Statement of the theorem

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 the theorem states that

:$lim\_\{varepsilon\; ightarrow\; 0^+\}\; int\_a^b\; frac\{f(x)\}\{xpm\; i\; varepsilon\},dx\; =\; mp\; i\; pi\; f(0)\; +\; mathcal\{P\}int\_a^b\; frac\{f(x)\}\{x\},$

where $mathcal\{P\}$ denotes the Cauchy principal value.

Proof of the theorem

A simple proof is as follows.

:$lim\_\{varepsilon\; ightarrow\; 0^+\}\; int\_a^b\; frac\{f(x)\}\{xpm\; i\; varepsilon\},dx\; =\; mp\; i\; pi\; lim\_\{varepsilon\; ightarrow\; 0^+\}\; int\_a^b\; frac\{varepsilon\}\{pi(x^2+varepsilon^2)\}f(x),dx\; +\; lim\_\{varepsilon\; ightarrow\; 0^+\}\; int\_a^b\; frac\{x^2\}\{x^2+varepsilon^2\}\; ,\; frac\{f(x)\}\{x\},\; dx.$

For the first term, we note that $varepsilon/(pi(x^2+varepsilon^2))$ is an approximate identity, and therefore approaches a Dirac delta function in the limit. Therefore, the first term equals $mp\; i\; pi\; f(0)$.

For the second term, we note that the factor $x^2/(x^2+varepsilon^2)$ 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

:$int\_\{-infty\}^infty\; int\_0^infty\; f(E)exp(-iEt),dt,\; dE,$

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.:

:$lim\_\{varepsilon\; ightarrow\; 0^+\}\; int\_\{-infty\}^infty\; int\_0^infty\; f(E)exp(-iEt-varepsilon\; t),dt,\; dE$

::$=\; i\; lim\_\{varepsilon\; ightarrow\; 0^+\}\; int\_\{-infty\}^infty\; frac\{f(E)\}\{E-ivarepsilon\},dE\; =\; -pi\; f(0)+i\; mathcal\{P\}int\_\{-infty\}^\{infty\}frac\{f(E)\}\{E\},dE,$where the latter step uses this theorem.

References

* Chapter 3.1.
* Appendix A, equation (A.19).

References that refer to the theorem by name

Most sources that use this theorem, as mentioned above, refer to it generically as "a well-known theorem" or some variant. Here are some sources that refer to the theorem by name:
* [http://books.google.com/books?id=FL_u_J05qVEC&pg=PA60&lpg=PA60&dq=%22sokhotsky+weierstrass%22&source=web&ots=8yD1GxfjRJ&sig=ghjn35OX-bcW1MFLd_oBLbWczgs&hl=en "Quantum Theory of Many-body Systems: Techniques and Applications" by Alexandre M. Zagoskin, p60]
* [http://arxiv.org/abs/nucl-th/0212062 Relativistic effects in proton-induced deuteron break-up at intermediate energies with forward emission of a fast proton pair]
* [http://books.google.com/books?id=hB84gN0fmh8C&pg=PA259&lpg=PA259&dq=sokhotsky-weierstrass&source=web&ots=RLL0oRmps3&sig=P8SfPatLaSYtt-N9iTvApvQg_pA&hl=en "Two-dimensional Coulomb Liquids and Solids", Yuriy Monarkha, p259]