Sokhatsky-Weierstrass theorem

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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.

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[edit] 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\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},

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

[edit] Proof of the theorem

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 \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.

[edit] 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\rightarrow 0^+} \int_{-\infty}^\infty \int_0^\infty f(E)\exp(-iEt-\varepsilon t)\,dt\, dE
= 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 this theorem.

[edit] 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).

[edit] 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:

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