# 索霍茨基－魏尔斯特拉斯定理

The Sokhatsky–Weierstrass 定理 (亦作Sokhotsky–Weierstrass 定理, Sokhotski–Plemelj formula,[1]Weierstrass theorem（勿与 various other theorems called the "Weierstrass theorem"混淆）是複分析中的一个定理，用于计算很多问题中出现的柯西主值。物理学问题中很多见，但鲜有其命名的引用。该定理源自Yulian Sokhotski, Karl WeierstrassJosip Plemelj

## 定理陈述

ƒ为定义在实线上的连续複值函数，ab为实常数，满足a < 0 < b。则

$\lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{f(x)}{x\pm i \varepsilon}\,dx = \mp i \pi f(0) + \mathcal{P}\int_a^b \frac{f(x)}{x}\, dx,$

## 定理证明

$\lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{f(x)}{x\pm i \varepsilon}\,dx = \mp i \pi \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{\varepsilon}{\pi(x^2+\varepsilon^2)}f(x)\,dx + \lim_{\varepsilon\rightarrow 0^+} \int_a^b \frac{x^2}{x^2+\varepsilon^2} \, \frac{f(x)}{x}\, dx.$

## 物理应用

$\int_{-\infty}^\infty \int_0^\infty f(E)\exp(-iEt)\,dt\, dE,$

$\lim_{\varepsilon\rightarrow 0^+} \int_{-\infty}^\infty \int_0^\infty f(E)\exp(-iEt-\varepsilon t)\,dt\, dE$
$= -i \lim_{\varepsilon\rightarrow 0^+} \int_{-\infty}^\infty \frac{f(E)}{E-i\varepsilon}\,dE = \pi f(0)-i \mathcal{P}\int_{-\infty}^{\infty}\frac{f(E)}{E}\,dE,$

## 参考文献

1. ^ Blanchard, Philippe; Brüning, Erwin. Mathematical Methods in Physics. Boston: Birkhauser. 2003. ISBN 0817642285. Example 3.3.1 4.
• Weinberg, Steven. The Quantum Theory of Fields, Volume 1: Foundations. Cambridge Univ. Press. 1995. ISBN 0-521-55001-7. Chapter 3.1.
• Merzbacher, Eugen. Quantum Mechanics. Wiley, John & Sons, Inc. 1998. ISBN 0-471-88702-1. Appendix A, equation (A.19).