# 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函数ab为实常数，满足a < 0 < b。则

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

## 定理证明

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

## 物理应用

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

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

## 参考文献

1. ^ Blanchard, Philippe; Brüning, Erwin. Mathematical Methods in Physics. Boston: Birkhauser. 2003. ISBN 0817642285. Example 3.3.1 4.