# 克罗内克δ函数

（重定向自克罗内克δ

${\displaystyle \delta _{ij}=\left\{{\begin{matrix}1&(i=j)\\0&(i\neq j)\end{matrix}}\right.\,\!}$

## 其它记法

${\displaystyle \delta _{ij}=[i=j]\,\!}$

${\displaystyle \delta _{i}=\left\{{\begin{matrix}1,&{\mbox{if }}i=0\\0,&{\mbox{if }}i\neq 0\end{matrix}}\right.\,\!}$

## 数字信号处理

${\displaystyle \delta [n]={\begin{cases}1,&n=0\\0,&n\neq 0\end{cases}}\,\!}$

## 性质

${\displaystyle \sum _{i=-\infty }^{\infty }\delta _{ij}a_{i}=a_{j}\,\!}$

${\displaystyle \int _{-\infty }^{\infty }\delta (x-y)f(x)dx=f(y)\,\!}$

### 线性代数中的应用

• 作为线性映射的单位矩阵。
• 迹数
• 内积 ${\displaystyle V^{*}\otimes V\to K\,\!}$
• 映射 ${\displaystyle K\to V^{*}\otimes V\,\!}$ ，将数量乘积表示为外积的形式。

## 廣義克羅內克函數

${\displaystyle \delta _{i_{1}i_{2}\dots i_{n}}^{j_{1}j_{2}\dots j_{n}}={\begin{bmatrix}\delta _{i_{1}}^{j_{1}}\delta _{i_{2}}^{j_{1}}&\cdots &\delta _{i_{n}}^{j_{1}}\\\delta _{i_{1}}^{j_{2}}\delta _{i_{2}}^{j_{2}}&\cdots &\delta _{i_{n}}^{j_{2}}\\\vdots &\ddots &\vdots \\\delta _{i_{1}}^{j_{n}}\delta _{i_{2}}^{j_{n}}&\cdots &\delta _{i_{n}}^{j_{n}}\\\end{bmatrix}}\,\!}$

• ${\displaystyle \delta _{imn}^{ijk}=\delta _{mn}^{jk}=\delta _{m}^{j}\delta _{n}^{k}-\delta _{n}^{j}\delta _{m}^{k}\,\!}$
• ${\displaystyle \delta _{ijm}^{ijk}=2\delta _{m}^{k}\,\!}$
• ${\displaystyle \delta _{ijk}^{ijk}=6\,\!}$
• ${\displaystyle \delta _{lmn}^{ijk}=\epsilon ^{ijk}\epsilon _{lmn}\,\!}$

• ${\displaystyle \delta _{i_{1}i_{2}\dots i_{n}}^{j_{1}j_{2}\dots j_{n}}=\epsilon ^{j_{1}j_{2}\dots j_{n}}\epsilon _{i_{1}i_{2}\dots i_{n}}\,\!}$
• ${\displaystyle \delta _{i_{1}i_{2}\dots i_{n}}^{12\dots n}=\epsilon _{i_{1}i_{2}\dots i_{n}}\,\!}$
• ${\displaystyle \delta _{i_{1}i_{2}\dots i_{n}}^{j_{1}j_{2}\dots j_{n}}T_{j_{1}j_{2}\dots j_{n}}=n!\ T_{i_{1}i_{2}\dots i_{n}}\,\!}$

## 积分表示

${\displaystyle \delta _{x,n}={\frac {1}{2\pi i}}\oint z^{x-n-1}dz\,\!}$

${\displaystyle \delta _{x,n}={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{i(x-n)\varphi }d\varphi \,\!}$

## 參考文獻

1. ^ Heinbockel, J. H., Introduction to Tensor Calculus and Continum Mechanics, Victoria, B.C. Canada: Trafford Publishing: pp. 14, 31, 2001, ISBN 1-55369-133-4